C* -Algebras : Banach Spaces (North-Holland Mathematical Library)

C*-ALGEBRAS

V O L U M E 1" B A N A C H SPACES

North-Holland Mathematical Library Board of Honorary Editors: M. Artin, H. Bass, J. Eells, W. Feit, P.J. Freyd, F.W. Gehring, H. Halberstam, L.V. H6rmander, J.H.B. Kemperman, W.A.J.Luxemburg, F. Peterson, I.M. Singer and A.C. Zaanen

Board of Advisory Editors." A. Bj6mer, R.H. Dijkgraaf, A. Dimca, A.S. Dow, J.J. Duistermaat, E. Looijenga, J.P. May, I. Moerdijk, S.M. Mori, J.P. Palis, A. Schrijver, J. Sj6strand, J.H.M. Steenbrink, F. Takens and J. van Mill

VOLUME 58

ELSEVIER Amsterdam- London- New York- Oxford- Paris - Shannon- Tokyo

C*-Algebras Volume l: Banach Spaces

Corneliu Constantinescu Departement Mathematik, ETH Ziirich CH-8092 Ziirich Switzerland

2001 ELSEVIER A m s t e r d a m - L o n d o n - New Y o r k - O x f o r d - Paris - Shannon- Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

9 2001 Elsevier Science B.V. All rights reserved.

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Preface

Functional analysis plays an important role in the program of studies at the Swiss Federal Institute of Technology. At present, courses entitled Functional Analysis I and II are taken during the fifth and sixth semester~ respectively. I have taught these courses several times and after a while typewritten lecture notes resulted that were distributed to the students. During the academic year 1987/88, I was fortunate enough to have an eager enthusiastic group of students t h a t I had already encountered previously in other lecture courses. These students wanted to learn more in the area and asked me to design a continuation of the courses. Accordlingly, I proceeded during the academic year, following, with a series of special lectures, Functional Analysis III and IV, for which I again distributed typewritten lecture notes. At the end I found that there had accumulated a mass of textual material, and I asked myself if I should not publish it in the form of a book. Unfortunately, I realized that the two special lecture series (they had been given only once) had been badly organized and contained material that should have been included in the first two portions. And so I came to the conclusion that I should write everything anew - and if at a l l - then preferably in English. Little did I realize what I was letting myself in for! The number of pages grew almost impercepetibly and at the end it had more than doubled. Aslo, the English language turned out to be a stumbling block for me; I would like to take this opportunity to thank Prof. Imre Bokor and Prof. Edgar Reich for their help in this regard. Above all I must thank Mrs. Barbara Aquilino, who wrote, first a WordMARC TM., and then a IbTF~ TM version with great competence, angelic patience, and utter devotion, in spite of illness. My thanks also go to the Swiss Federal Institute of Technology that generously provided the infrastructure for this extensive enterprise and to my colleagues who showed their understanding for it.

Corneliu Constantinescu

This Page Intentionally Left Blank

vii

Table of Contents of Volume 1

Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some Notation and Terminology 1

Banach Spaces 1.1

1.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

N o r m e d Spaces

. . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.1

General Results . . . . . . . . . . . . . . . . . . . . . . .

1.1.2

Some Standard Examples

. . . . . . . . . . . . . . . . .

1 7 7 7 12

1.1.3

Minkowski's Theorem . . . . . . . . . . . . . . . . . . . .

31

1.1.4

L o c a l l y C o m p a c t N o r m e d Spaces

35

1.1.5

P r o d u c t s of N o r m e d Spaces

. . . . . . . . . . . . . . . .

37

1.1.6

S u m m a b l e Families . . . . . . . . . . . . . . . . . . . . .

40

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Operators 1.2.1

. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

General Results

. . . . . . . . . . . . . . . . . . . . . .

61 61

1.2.2

Standard Examples . . . . . . . . . . . . . . . . . . . . .

74

1.2.3

Infinite M a t r i c e s

. . . . . . . . . . . . . . . . . . . . . .

92

1.2.4

Q u o t i e n t Spaces . . . . . . . . . . . . . . . . . . . . . . .

113

1.2.5

Complemented Subspaces

1.2.6

T h e T o p o l o g y of Pointwise C o n v e r g e n c e

. . . . . . . . . . . . . . . . . .........

123 134

1.2.7

C o n v e x Sets . . . . . . . . . . . . . . . . . . . . . . . . .

138

1.2.8

The Alaoglu-Bourbaki Theorem . . . . . . . . . . . . . .

148

Bilinear Maps . . . . . . . . . . . . . . . . . . . . . . . .

150

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

153

1.2.9

1.3

. . . . . . . . . . . . . . . . . . . .

xix

The Hahn-Banach Theorem 1.3.1

. . . . . . . . . . . . . . . . . . . .

159

The Banach Theorem . . . . . . . . . . . . . . . . . . . .

159

1.3.2

E x a m p l e s in M e a s u r e T h e o r y

. . . . . . . . . . . . . . .

171

1.3.3

The Hahn-Banach Theorem

. . . . . . . . . . . . . . . .

180

1.3.4

T h e T r a n s p o s e of an O p e r a t o r . . . . . . . . . . . . . . .

191

viii

1.4

1.5

1.6

Table of Contents

1.3.5

P o l a r Sets

1.3.6

The Bidual

. . . . . . . . . . . . . . . . . . . . . . . . . .

199

. . . . . . . . . . . . . . . . . . . . . . . . .

211

1.3.7

The Krein-Smulian Theorem

. . . . . . . . . . . . . . .

228

1.3.8

Reflexive S p a c e s . . . . . . . . . . . . . . . . . . . . . . .

240

1.3.9

C o m p l e t i o n of N o r m e d Spaces . . . . . . . . . . . . . . .

245

1.3.10

Analytic Functions

246

. . . . . . . . . . . . . . . . . . . . .

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

254

A p p l i c a t i o n s of B a i r e ' s T h e o r e m . . . . . . . . . . . . . . . . . .

256

1.4.1

The Banach-Steinhaus Theorem . . . . . . . . . . . . . .

256

1.4.2

Open Mapping Principle

. . . . . . . . . . . . . . . . . .

264

E x ercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

280

Banach Categories

. . . . . . . . . . . . . . . . . . . . . . . . .

281

1.5.1

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . .

281

1.5.2

Functors . . . . . . . . . . . . . . . . . . . . . . . . . . .

288

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

308

1.6.1

General Results . . . . . . . . . . . . . . . . . . . . . . .

308

1.6.2

Examples

. . . . . . . . . . . . . . . . . . . . . . . . . .

322

O r d e r e d B a n a c h spaces . . . . . . . . . . . . . . . . . . . . . . .

334

1.7.1

O r d e r e d n o r m e d spaces . . . . . . . . . . . . . . . . . . .

334

1.7.2

Order Continuity

. . . . . . . . . . . . . . . . . . . . . .

340

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

357

Subject Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

359

Symbol Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

371

1.7

Nuclear Maps

Name Index

ix

C o n t e n t s of All V o l u m e s

T a b l e o f C o n t e n t s of V o l u m e 1

Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some Notation and Terminology B a n a c h Spaces 1.1

1.2

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

N o r m e d Spaces

. . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.1

General Results . . . . . . . . . . . . . . . . . . . . . . .

1.1.2

Some S t a n d a r d Examples

. . . . . . . . . . . . . . . . .

1 7 7 7 12

1.1.3

Minkowski's Theorem . . . . . . . . . . . . . . . . . . . .

31

1.1.4

L o c a l l y C o m p a c t N o r m e d Spaces

. . . . . . . . . . . . .

35

1.1.5

P r o d u c t s of N o r m e d Spaces

. . . . . . . . . . . . . . . .

37

1.1.6

S u m m a b l e Families . . . . . . . . . . . . . . . . . . . . .

40

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Operators 1.2.1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

General Results

61

. . . . . . . . . . . . . . . . . . . . . .

61

1.2.2

Standard Examples . . . . . . . . . . . . . . . . . . . . .

74

1.2.3

Infinite M a t r i c e s

. . . . . . . . . . . . . . . . . . . . . .

92

1.2.4

Q u o t i e n t Spaces . . . . . . . . . . . . . . . . . . . . . . .

113

1.2.5

Complemented Subspaces

123

1.2.6

T h e T o p o l o g y of P o i n t w i s e C o n v e r g e n c e

. . . . . . . . . . . . . . . . . .........

134

1.2.7

C o n v e x Sets . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.8

The Alaoglu-Bourbaki Theorem . . . . . . . . . . . . . .

148

1.2.9

Bilinear Maps . . . . . . . . . . . . . . . . . . . . . . . .

150

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3

xix

The Hahn-Banach Theorem

. . . . . . . . . . . . . . . . . . . .

138

153 159

1.3.1

The Banach Theorem . . . . . . . . . . . . . . . . . . . .

159

1.3.2

E x a m p l e s in M e a s u r e T h e o r y

. . . . . . . . . . . . . . .

171

1.3.3

The Hahn-Banach Theorem

. . . . . . . . . . . . . . . .

180

1.3.4

T h e T r a n s p o s e of a n O p e r a t o r . . . . . . . . . . . . . . .

191

x

1.4

Table of Contents

1.3.5

Polar Sets

1.3.6

The Bidual

. . . . . . . . . . . . . . . . . . . . . . . . . .

199

. . . . . . . . . . . . . . . . . . . . . . . . .

211

1.3.7

The Krein-Smulian

1.3.8

Reflexive Spaces . . . . . . . . . . . . . . . . . . . . . . .

Theorem

1.3.9

C o m p l e t i o n of N o r m e d Spaces

1.3.10

Analytic Functions

. . . . . . . . . . . . . . .

1.6

1.7

240

. . . . . . . . . . . . . . .

245

. . . . . . . . . . . . . . . . . . . . .

246

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

254

A p p l i c a t i o n s of Baire's T h e o r e m

. . . . . . . . . . . . . . . . . .

256

Theorem . . . . . . . . . . . . . .

256

1.4.1

The Banach-Steinhaus

1.4.2

Open Mapping Principle

. . . . . . . . . . . . . . . . . .

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5

228

Banach Categories

264 280

. . . . . . . . . . . . . . . . . . . . . . . . .

281

1.5.1

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . .

281

1.5.2

Functors

Nuclear Maps

. . . . . . . . . . . . . . . . . . . . . . . . . . .

288

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

308

1.6.1

General Results

1.6.2

Examples

. . . . . . . . . . . . . . . . . . . . . . .

308

. . . . . . . . . . . . . . . . . . . . . . . . . .

322

Ordered Banach spaces . . . . . . . . . . . . . . . . . . . . . . .

334

1.7.1

Ordered normed spaces . . . . . . . . . . . . . . . . . . .

334

1.7.2

Order Continuity

340

Name Index

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

357

Subject Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

359

Symbol Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

371

xi

Table of Contents of Volume 2

Introduction 2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Banach Algebras 2.1

2.2

2.3

2.4

Algebras

xix

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.1.1

General Results

2.1.2

Invertible Elements

2.1.3

The Spectrum

2.1.4

Standard

2.1.5

C o m p l e x i f i c a t i o n of A l g e b r a s . . . . . . . . . . . . . . . .

51

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

Normed Algebras

. . . . . . . . . . . . . . . . . . . . . . .

3

. . . . . . . . . . . . . . . . . . . . .

13

. . . . . . . . . . . . . . . . . . . . . . . .

17

Examples

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

2.2.1

General Results

2.2.2

The Standard

2.2.3

The Exponential

2.2.4

Invertible E l e m e n t s of U n i t a l B a n a c h A l g e b r a s . . . . . .

2.2.5

The Theorems

2.2.6

Poles of R e s o l v e n t s

2.2.7

Modules

69

. . . . . . . . . . . . . . . . . . . . . . .

Examples

. . . . . . . . . . . . . . . . . .

Function and the Neumann

Series

. . .

of R i e s z a n d G e l f a n d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

32

69 82 114 125 153 161

.....

174

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

197

Involutive Banach Algebras

. . . . . . . . . . . . . . . . . . . .

201

. . . . . . . . . . . . . . . . . . . . .

201

2.3.1

Involutive Algebras

2.3.2

Involutive Banach Algebras

2.3.3

Sesquilinear Forms

2.3.4

Positive Linear Forms

. . . . . . . . . . . . . . . . . . .

287

2.3.5

The State Space . . . . . . . . . . . . . . . . . . . . . . .

305

2.3.6

Involutive Modules

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

241 275

. . . . . . . . . . . . . . . . . . . . .

322

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

328

Gelfand Algebras

. . . . . . . . . . . . . . . . . . . . . . . . . .

331

2.4.1

The Gelfand Transform . . . . . . . . . . . . . . . . . . .

331

2.4.2

Involutive Gelfand Algebras

343

. . . . . . . . . . . . . . . .

xii

Table o] Contents

2.4.3

Examples

2.4.4

Locally Compact Additive Groups .............

. . . . . . . . . . . . . . . . . . . . . . . . . .

365

2.4.5

Examples

378

2.4.6

The Fourier Transform

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Compact Operators 3.1

3.2

. . . . . . . . . . . . . . . . . . . . . . . . . . .

The General Theory

358

390 396 399

. . . . . . . . . . . . . . . . . . . . . . . .

399

3.1.1

General Results . . . . . . . . . . . . . . . . . . . . . . .

399

3.1.2

Examples

419

3.1.3

Fredholm Operators

3.1.4

Point Spectrum

3.1.5

S p e c t r u m of a C o m p a c t O p e r a t o r

3.1.6

Integral Operators

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .............

. . . . . . . . . . . . . . . . . . . . .

437 468 477 489

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

517

L i n e a r Differential E q u a t i o n s . . . . . . . . . . . . . . . . . . . .

518

3.2.1

B o u n d a r y Value P r o b l e m s for Differential E q u a t i o n s . . .

518

3.2.2

Supplementary Results . . . . . . . . . . . . . . . . . . .

530

3.2.3

L i n e a r P a r t i a l Differential E q u a t i o n s

...........

549

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

563

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

565

Subject Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

568

Symbol Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

588

Name Index

xiii

T a b l e o f C o n t e n t s of V o l u m e 3

Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix

C*-Algebras

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

4.1

4.2

4.3

4.4

The General Theory

. . . . . . . . . . . . . . . . . . . . . . . .

3

4.1.1

General Results . . . . . . . . . . . . . . . . . . . . . . .

4

4.1.2

T h e S y m m e t r y of C * - A l g e b r a . . . . . . . . . . . . . . .

30

4.1.3

F u n c t i o n a l c a l c u l u s in C * - A l g e b r a s

56

4.1.4

T h e T h e o r e m of F u g l e d e - P u t n a m

............ .............

75

The Order Relation . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1

Definition and General Properties

4.2.3

Examples

4.2.4

P o w e r s of P o s i t i v e E l e m e n t s

4.2.5

The Modulus

4.2.6

Ideals a n d Q u o t i e n t s of C * - A l g e b r a s

4.2.7

T h e O r d e r e d Set of O r t h o g o n a l P r o j e c t i o n s

4.2.8

Approximate Unit

92

.............

92

. . . . . . . . . . . . . . . . . . . . . . . . . .

116

. . . . . . . . . . . . . . . .

123

. . . . . . . . . . . . . . . . . . . . . . . .

143

...........

150

.......

162

. . . . . . . . . . . . . . . . . . . . .

178

S u p p l e m e n t a r y R e s u l t s on C * - A l g e b r a s . . . . . . . . . . . . . .

208

4.3.1

. . . . . . . . . . . . . . . .

208

. . . . . . . . . . . . . . .

215

The Exterior Multiplication

4.3.2

Order Complete C*-Algebras

4.3.3

The Carrier

4.3.4

Hereditary C*-Subalgebras

. . . . . . . . . . . . . . . .

263

4.3.5

Simple C*-algebras . . . . . . . . . . . . . . . . . . . . .

276

4.3.6

Supplementary Results Concerning Complexification

W*-Algebras 4.4.1

. . . . . . . . . . . . . . . . . . . . . . . . .

243

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

General Properties

297

. . . . . . . . . . . . . . . . . . . . .

297

4.4.2

F as an E - s u b m o d u l e

. . . . . . . . . . . . . . .

309

4.4.3

Polar Representation

. . . . . . . . . . . . . . . . . . . .

335

4.4.4

W*-Homomorphisms

. . . . . . . . . . . . . . . . . . .

Name Index

of E '

286

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

361 385

xiv

Table of Contents

Subject Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

388

Symbol Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

411

xv

Table of Contents of Volume 4

Introduction

...............................

5 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Pre-Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Hilbert sums . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Orthogonal Projections of Hilbert Space . . . . . . . . . . . . . 5.2.1 Projections onto Convex Sets . . . . . . . . . . . . . . . 5.2.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Orthogonal Projections . . . . . . . . . . . . . . . . . . . 5.2.4 Mean Ergodic Theorems . . . . . . . . . . . . . . . . . . 5.2.5 The Frkchet-Riesz Theorem . . . . . . . . . . . . . . . . 5.3 Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Supplementary Results . . . . . . . . . . . . . . . . . . . 5.3.3 Selfadjoint Operators . . . . . . . . . . . . . . . . . . . . 5.3.4 Normal Operators . . . . . . . . . . . . . . . . . . . . . . 5.4 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Cyclic Representation . . . . . . . . . . . . . . . . . . . 5.4.2 General Representations . . . . . . . . . . . . . . . . . . 5.4.3 Example of Representations . . . . . . . . . . . . . . . . 5.5 Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Hilbert Dimension . . . . . . . . . . . . . . . . . . . . . 5.5.3 Standard Examples . . . . . . . . . . . . . . . . . . . . . 5.5.4 The Fourier-Plancherel Operator . . . . . . . . . . . . . 5.5.5 Operators and Orthonormal Bases . . . . . . . . . . . . 5.5.6 Self-normal Compact Operators . . . . . . . . . . . . . . 5.5.7 Examples of Real C-Algebras . . . . . . . . . . . . . .

XiX

3 3 3 14 19 24 24 29 33 54 63 72 72 86 108 123 130 130 146 156 166 166 191 206 218 223 243 258

Table o] Contents

xvi

5.6

Hilbert right C*-Modules 5.6.1

. . . . . . . . . . . . . . . . . . . . .

Some General Results

286

. . . . . . . . . . . . . . . . . . .

286

. . . . . . . . . . . . . . . . . . . . . . . . .

310

5.6.2

Self-duality

5.6.3

Von Neumann

5.6.4

Examples

. . . . . . . . . . . . . . . . . . . . . . . . . .

373

5.6.5

JCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

430

5.6.6

Matrices over C * - a l g e b r a s

. . . . . . . . . . . . . . . . .

477

5.6.7

Type I W*-algebras

. . . . . . . . . . . . . . . . . . . .

515

Name Index

right W*-modules

. . . . . . . . . . . . .

341

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

535

Subject Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

539

Symbol Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

567

xvii

Table of Contents of Volume 5

Introduction 6

...............................

Selected Chapters of C'-Algebras . . . . . . . . . . . . . . . . . . . 6.1 LP.Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Characteristic Families of Eigenvalues . . . . . . . . . . . 6.1.2 Characteristic Sequences . . . . . . . . . . . . . . . . . . 6.1.3 Properties of the CP-spaces . . . . . . . . . . . . . . . . 6.1.4 Hilbert-Schmidt Operators . . . . . . . . . . . . . . . . . 6.1.5 TheTrace . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.6 Duals of Cp-spaces . . . . . . . . . . . . . . . . . . . . . 6.1.7 Exterior Multiplication and Cp-Spaces . . . . . . . . . . 6.1.8 The Canonical Projection of the Tridual of K . . . . . . 6.1.9 Integral Operators on Hilbert Spaces . . . . . . . . . . . 6.2 Selfadjoint Linear Differential Equations . . . . . . . . . . . . . 6.2.1 Selfadjoint Boundary Value Problems . . . . . . . . . . . 6.2.2 The Regular Sturm-Liouville Theory . . . . . . . . . . . 6.2.3 Selfadjoint Linear Differential Equations on T . . . . . . 6.2.4 Associated Parabolic and Hyperbolic Evolution Equations 6.2.5 Selfadjoint Linear Partial Differential Equations . . . . . 6.2.6 Associated Parabolic and Hyperbolic Evolution Equations 6.3 Von Neumann Algebras . . . . . . . . . . . . . . . . . . . . . . 6.3.1 6.3.2 6.3.3 6.3.4 6.3.5 6.3.6 6.3.7 6.3.8 6.3.9

The Strong Topology . . . . . . . . . . . . . . . . . . . . Bidual of a C*-algebra . . . . . . . . . . . . . . . . . . . Extension of the Functional Calculus . . . . . . . . . . . Von Neumann- Algebras . . . . . . . . . . . . . . . . . . The Commutants . . . . . . . . . . . . . . . . . . . . . . Irreducible Representations . . . . . . . . . . . . . . . . Commutative von Neumann Algebras . . . . . . . . . . . Representations of W*-Algebras . . . . . . . . . . . . . . Finite-dimensional C*-algebras . . . . . . . . . . . . . .

xix

3 3 3 10 21 46 56 72 79 102 116 124 125 139 150 153 184 192 202 203 218 263 283 293 299 320 325 334

xviii

Table of Contents

6.3.10 A generalization . . . . . . . . . . . . . . . . . . . . . . . 7

C * - a l g e b r a s G e n e r a t e d by Groups 7.1

7.2

...................

P r o j e c t i v e Representations of Groups

...............

. . . . . . . . . . . . . . . . . . . . . . .

355 369 369

7.1.1

Schur functions

7.1.2

Projective Representations . . . . . . . . . . . . . . . . .

7.1.3

S u p p l e m e n t a r y Results . . . . . . . . . . . . . . . . . . .

431

7.1.4

Examples

. . . . . . . . . . . . . . . . . . . . . . . . . .

466

Clifford Algebras

. . . . . . . . . . . . . . . . . . . . . . . . . .

492

404

7.2.1

G e n e r a l Clifford Algebras

7.2.2

C~.p,q . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

518

7.2.3

C~(IN)

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

538

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

559

Subject Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

563

Symbol Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

592

N a m e Index

.................

369

492

xix

Introduction This book has evolved from the lecture course on Functional Analysis I had given several times at the ETH. The text has a strict logical order, in the style of "Definiton- Theorem - P r o o f - E x a m p l e - Exercises". The proofs are rather thorough and there are many examples. The first part of the book (the first three chapters, resp. the first two volumes) is devoted to the theory of Banach spaces in the most general sense of the term. The purpose of the first chapter (resp. first volume) is to introduce those results on Banach spaces which are used later or which are closely connected with the book. It therefore only contains a small part of the theory, and several results are stated (and proved) in a diluted form. The second chapter (which together with Chapter 3 makes the second volume) deals with Banach algebras (and involutive Banach algebras), which constitute the main topic of the first part of the book. The third chapter deals with compact operators on Banach spaces and linear (ordinary and partial) differential equations- applications of the theory of Banach algebras. The second part of the book (the last four chapters, resp. the last three volumes) is devoted to the theory of Hilbert spaces, once again in the general sense of the term. It begins with a chapter (Chapter 4, resp. Volume 3) on the theory of C*-algebras and W*-algebras which are essentially the focus of the book. Chapter 5 (resp. Volume 4) treats Hilbert spaces for which we had no need earlier. It contains the representation theorems, i.e. the theorems on isometries between abstract C*-algebras and the concrete C*-algebras of operators on Hilbert spaces. Chapter 6 (which together with Chapter 7 makes Volume 5) presents the theory of/:P-spaces of operators, its application to the self-adjoint linear (ordinary and partial) differential equations, and the von Neumann algebras. Finally, Chapter 7 presents examples of C*-algebras defined with the aid of groups, in particular the Clifford algebras. Many important domains of C*-algebras are ignored in the present book. It should be emphasized that the whole theory is constructed in parallel for the real and for the complex numbers, i.e. the C*-algebras are real or complex. In addition to the above (vertical) structure of the book, there is also a second (horizontal) division. It consists of a main strand, eight branches, and additional material. The results belonging to the main strand are marked with (0). Logically speaking, a reader could restrict himself/herself to these and ignore the rest. Results on the eight subsidiary branches are marked with (1), (2), (3), (4), (5), (6), (7), and (8). The key is

xx

Introduction

1. 2. 3. 4. 5. 6. 7. 8.

Infinite Matrices Banach Categories Nuclear Maps Locally Compact Groups Differential Equations Laurent Series Clifford Algebras Hilbert C*-Modules

These are (logically) independent of each other, but all depend on the main strand. Finally, the results which belong to the additional material have no marking and - from a logical perspective - may be ignored. So the reader can shorten for himself/herself this very long book using the above marks. Also, since the proofs are given with almost all references, it is possible to get into the book at any level and not to read it linearly. We assume that the reader is familiar with classical analysis and has rudimentary knowledge of set theory, linear algebra, point-set topology, and integration theory. The book addresses itself mainly to mathematicians, or to physicists interested in C*-algebras. I would like to apologize for any omissions in citations occasioned by the fact that my acquaintance with the history of functional analysis is~ unfortunately, very restricted. For this history we recommand the following texts. BIRKHOFF, G. and KREYSZIG, E., The Establishment of Functional Analysis, Historia Mathematica 11 (1984), 258-321. 2. BOURBAKI, N., Elements of the History of Mathematics. (21. Topological Vector Spaces), Springer-Verlag (1994). 3. DIEUDONNt~, J., History of Functional Analysis, North-Holland (1981). 4. DIEUDONNI~, J., A Panorama of Pure Mathematics (Chapter C III: Spectral Theory of Operators), Academic Press (1982). HEUSER, H., Funktionalanalysis, 2. Auflage (Kapitel XIX: Ein Blick auf die werdende Functionalanalysis), Teubner (1986), 3. Auflage (1992). KADISON, R.V., Operator Algebras, the First Forty Years, in: Proceedings of Symposia in Pure Mathematics 38 I (1982), 1-18. MONNA, A.F., Functional Analysis in Historical Perspective~ John Whiley & Sons (1973).

Introduction

xxi

8. STEEN, L.A., Highlights in the History of Spectral Theory, Amer. Math. Monthly 80 (1973), 359-382. There is no shortage of excellent books on C*-algebras. Nevertheless, we hope that this book will be also of some utility to the mathematics commutity.

This Page Intentionally Left Blank

Some N o t a t i o n and Terminology We use in this book the notation and terminology which are usual in the current m a t h e m a t i c a l literature. In the following list we present some of those for which we felt that difficulties in interpretation may arise. Any set theory (with the axiom of choise or equivalently with Zorn's Lemma) will do for the present book. 3 and V denote "there exists" and "for all", respectively; 3! means "there exists uniquely". We write iff for "if and only if". On special occasions (appearing very seldom) we choose the axiomatic setting of von Neumann: we call class a collection of sets (which need not be itself a set) and we define an ordinal number ~ as the set of ordinal numbers 7/ strictly smaller than ~, i.e.

(0 := 0). A cardinal number is the smallest ordinal number having a given cardinality; we denote for every set T by C a r d T its cardinal number, i.e. the cardinal number with the same cardinality as T . If P is a proposition and x a variable (which may occur in P ) , then

{x IF(x)} denotes the class of x for which P(x) holds. If in addition X is a set, then we put

{x e x lP(x)} : - {x Ix e x and P ( x ) } . If A and B are sets, then

A\B := {x e d lx ~ B}, A A B := (A\B)U (B\A), dxB:={(x,y)

lxed

and y e B } .

A partition of a set X is a set of pairwise disjoint nonempty subsets of X the union of which is equal to X . A function or a map is a triple f := (X, Y, F) (denoted also by f : X -+ Y ), where X , Y are sets and F is a subset of X x Y such that

x E X ==~ 3!y~ C Y, (x, yz) C F. X, Y, and F are called the domain, the range of values (or codomain), and the graph of f , respectively. We set then f(x) := y~

for all x E X and

f ( A ) := {y E Y I ::Ix E A , f ( x ) = y}, -1

) E B},

f(B):={xEXIf(x -1

-1

f (Y):= f ({Y}) for all A c X , B

C Y , and y E Y. We call f a m a p o f

X into Y. If f , g

are maps of X into Y, then we set {f = g} := {x E

X lf(x) = g ( x ) } ,

{f -~ g} : : {x e X I f ( x ) ~ g ( x ) } . If T is a term and x a variable (which may occur in T ) and X, Y are sets such that

x EX ~

T(x) E Y ,

then we denote the map f := (X,Y, {(x,y) E X x Y IY = T(x)}) by

f : X -~ Y,

x,

>T ( x ) .

If f : X --+ Y is a map and Z is a subset of X , then the restriction of f to Z (denoted f i Z ) i s

the map

z--~y,

x,

~f(z).

The map f : X --+ Y is called injective (surjective), if (~, y e X ,

f(x) = I(Y)) ~

x = y

( / ( x ) = Y). The expression "f is a map of X onto Y" means f is a surjective map of X into Y. f is called bijective if it is simultaneously injective and surjective, in which case we set f-l

:y

; X,

f (x) ,

>z

and call f - 1 the inverse of f . If Y is a set and X is a subset of Y , then the inclusion map X --+ Y is the map X

>Y,

x~

>x;

if X = Y then we may call the inclusion map X -+ Y the identity map of Y . If X, Y, Z are sets and

f :X

~Y,

g:Y

>Z ,

x.

>g ( f ( x ) )

then we put

go f : X - - + Z ,

and call this map the composition of f and g. If X, Y, Z are sets and

f:XxY

>Z,

then we put

f ( a , . ) : Y----+ Z ,

y,

> f(a,y),

>Z,

x,

~f(x,b)

f(-,b):X for all a E X

and b E Y .

A family (X)~Er (indexed by I ) is in fact the m a p c ~ x~ defined on the set I for which the range of values (codomain) is not specified. Any set X defines the canonical family (x)xEx. If (X~),Er is a family of sets, then we put

1-I x~ .= ((x,),,~ !~ E t ~

x, E x~}

and call 11 X~ the product of the family (X~)~ei. tEI

Let X be a set. An equivalence relation on X is a binary relation ~ on X such t h a t we have for all x, y, z E X " Xt,,.,X~

x~y::~y,',-,x, ( x , ' ~ y and y ~ z )

~x~z.

An equivalence class of the equivalence relation ~ is a nonmepty subset A of X such t h a t

x , y E A ==~ x ~ y , (x c A , y C X , x ~ y) :==v y E A . For every x C X , the set {y C X I x ~ y} is an equivalence classe of ,-~ called the equivalence class of x (with respect to ~ ). The set of equivalence classes of is a partition of X which is denoted by X / ~ .

The m a p X --+ X / ~

which

sends every x C X into its equivalence class is called the quotient map. A free ultrafilter on a set X is an ultrafilter on X possessing no one-point sets. If X is an infinite set then the filter on X , {X\AIA

finite set}

is called the filter of cofinite subsets of A. A totally ordered set is an ordered set X such that for all x , y E X either x <_ y or y _< x . IN, 2Z,~, IR, and G denote the sets of natural numbers, integers, rational numbers, real numbers, and complex numbers, respectively (we do not include 0 among the natural numbers). A sequence is a family indexed by IN. If rn, n E and p c IN, then m = n

(modp)

means m - n

ETZ,

P (in words: m equal n modulo p ) . If

]R-= IRu {-oo, oo} and a, fl c IR, then

]~,9[- { z e ~ l ~ < ~ < Z } , [~,9]- {~e ~I~_<~_
{f > a} := {x e X l f ( x ) > a}. An additive group is a commutative group for which the composition law and the neutral element are denoted by + and 0, respectively. If F

is a

commutative field and s,t are variables, then F[t] and F[s,t] denote the set of polynomials in the variables t and s and t, repsectively, with coefficients in F . The sets of polynomial in more variables (which will appear in the exercises only) will be denoted similarly. If A is a square matrix over ~ , then Det A denotes the determinant of A. If T is a topological space and A is a subset of T , then A and ~l denote the closure and the interior of A, respectively; their elements are called adherent points (or points of adherence) and interior points of A, respectively. If f is a function defined on T with values in a vector space then the support or the carrier of f (denoted by Supp f ) is the set { f ~ 0}. The topological cardinality of T is the smallest cardinal number R such that there is a dense set of T of cardinality b~. Let T be a Hausdorff topological space, J~ the set of compact sets of T , and iR the set of relatively compact Borel sets of T . A Radon measure # on T is a (real or complex) measure # on 9q, such t h a t for every c > 0 there is a K 'C ~ , K C A, such that

I#(L) - #(A)I < c for every L E ~ with K C L c A. A set A of 9~ is called a p-null set if

#(B) = 0 for all B c iR, B C A. A #-null set is a subset A of T such that for every K C ~ , there is a p-null set B E iR containing A V1K . The carrier (or support) of it (denoted Supp #) is the smallest closed set F of T such that T \ F is a p-null set. The absolute value of # (denoted I#1) is the smallest Radon measure ~ on T such t h a t Ip(A)l _< L,(A) for all A c 9l. A step function on T with respect to ~ is a real or complex -1

function f on T such that f (T) is finite and f (a) E 9~ for all c~ =/- 0.

This Page Intentionally Left Blank

1. B a n a c h Spaces

The theory of Banach spaces rears its head even when one wishes to devote attention exclusively to Hilbert spaces, for the set of operators on a Hilbert space only forms a Banach space and not a Hilbert space. Hence the first chapter of our book studies Banach spaces, focussing on those aspects of the theory which find application below.

1.1

Normed

Spaces

The basic properties of normed spaces are presented, with emphasis on finitedimensional spaces since the finite-dimensional subspaces play an important role in the theory of compact operators. The principal results are Minkowski's Theorem and Riesz's Theorem.

1.1.1 G e n e r a l R e s u l t s /

Convention

1.1.1.1

[

0

\

)

We use IK throughout this book to denote either

lit, the field of real numbers, or ~ , the field of complex numbers. The elements of ]K are called scalars. All vector spaces are vector spaces over IK. A linear f o r m on the vector space E is a linear map of E into ]K. The vector space of all linear forms on E is called the algebraic dual o f

E.

Given a scalar,

E ]K, we use re c~ to denote its real part, imc~ to denote its i m a g i n a r y part, 9: rec~ - iimc~ to denote the conjugate o f ~ , and [oe[ := v/(re c~)2 + (im c~)2 to denote the absolute value o f o~.

8

1. Banach Spaces

If s,t

are variables, then

lK[t] and IK[s,t] denote the set of polynomials

in the variable t and s and t , respectively, with coefficients in IK.

Definition 1.1.1.2

( 0 )

A seminorm

on the v e c t o r space E is a m a p

>IR+

p: E such that for every x, y C E and a c lK p ( x + y) <_ p(x) +

p(y),

(Triangle Inequality)

p(~x) = Io~lp(x). If, moreover, p(x) -- O

only if x - - O ,

then p is said to be a n o r m on E , in which case it is customary to write

Ilxll

for p(~)

and to call the scalar lixll the n o r m of ~. We d~fn~

E ~ : - { x e E l l x l <1}. E # is called the unit ball o f E .

Two norms p, q on a vector space are said to be equivalent if 1 - p <_ q <_ c~p o~

for some ~ > O. A n o r m e d space is a vector space endowed with a norm, which we usually

d~not~ by If.ll. Given E a normed space, the map

E • E

~ ~+,

(x,y) ,

) l l ~ - yll

is a metric on E , called the c a n o n i c a l m e t r i c o f E . Unless otherwise specified, we take every normed space to be endowed with its canonical metric. Its topology is called the n o r m topology. I f this metric is complete, then we say that the norm is c o m p l e t e or that E

is complete. In this case we call E

space. A subset A of E is called bounded if

a Banach

1.1 Normed Spaces

9

~ u p Ilxll < o ~ .

~A

A sequence (Xn)neIN in E is called bounded if the subset {Xn I n e IN} of E is bounded. Given a set T , a map x : T --+ E is called bounded if x ( T ) a bounded set of E . Let F be a vector subspace of E . norm of E to F is a norm on F

is

The restriction of the

(the induced n o r m on F ); F

endowed

with this norm is called subspace o f E . Normed spaces over IK are called real n o r m e d spaces if ]K = ]R and c o m p l e x n o r m e d spaces if ]K = (~. I f they are complete, we call them real (rasp. c o m p l e x ) B a n a c h spaces. Given a metric space T , we define for each t C T and ce > O, Us(t) : - u T ( t ) : = {s c T Id(s, t) < c~}, where d denotes the metric of T . Remark.

1. In 1908 M. Fr(~chet and E. Schmidt introduced a norm on t~2

(Example 1.1.2.5), w i t h o u t calling it a norm. F. Riesz did the same for C([a, b]) in 1916 (Example 1.1.2.4), and E. Helly for spaces of sequences in 1921. T h e general notion of a norm on a vector space was introduced by H. Hahn and S. Banach in 1923. The notion of b o u n d e d set of a normed space was i n t r o d u c e d by A. Kolmogoroff and J. von N e u m a n n in 1935. 2. Let F be a subspace of the normed space E . If F is complete, t h e n it is a closed set of E . T h e converse holds whenever E is complete. 3. The seminorm was introduced by H. Minkowski in 1896.

Proposition 1.1.1.3 ( 0 )

Let p be a s e m i n o r m on the vector space E .

Then p(o) = o

and Ip(x) - p ( p ) l _ p ( x - y)

for every x , y E E . We have p(o) = p(o.o)

From

-

Iolp(o) = o.

10

1. Banach Spaces

p(~) = p((x - y) + y) _< p(x - v) + p(y)

we get p(x) - p ( y ) < p ( ~ - y )

We deduce p ( v ) - p ( x ) <__ v ( v - ~) = p ( ~ - v)

and so Ip(~) - p ( y ) l _

Corollary 1.1.1.4

p(x - y).

m

( 0 ) Given a normed space E , the map )~+,

E

~,

>llxll

is uniformly continuous and the set E # is closed,

P r o p o s i t i o n 1.1.1.5

m

( 5 ) Let (x~)~ei be a finite family in the real (complex)

normed space E . Put

o sup{ and let (a~)~el be a family in IK such that

supla~[ < 1. tEI

Then

Step1

<4~

(I

_< 2c~

{~I~EI}c1R+-+

~x~

_<~

We may assume that I - {k E IN I k _< n} for some n c I N and that (c~k)keI is decreasing. Then ~ k--1

_< ~ ( ~ k

-

ii

~(~-~+,)

xz+~n

k--1

i--1

- ~k+~)

~i

+ ~

k=l n-1

i=1

___ ~ ( ~ k

- ~k+,)~ + ~ . ~ _< ~ .

k=l

x, i=1

~ i=1

_<

<

1.1 Normed Spaces

Step 2

11

_~ 2a

We put J'={eEIla~_>O},

K'-{~EI

la~ <0}.

By Step 1,

~/OQxe ~E

EJ

<_ a + a - -

2a.

By Step 2,

II~~FI-~II~,r~,~ 9 ~,-~~-4o,~1 imo~,x~ll

m

12

1. Banach Spaces

1.1.2 S o m e S t a n d a r d E x a m p l e s D e f i n i t i o n 1.1.2.1 ( 0 ) Given sets E and T , we denote by E T the set of all maps of T into E . If E is a vector space (additive group), then E T becomes a vector space (additive group) when the operations are defined pointwise. If E is an additive group, we define E (r) := {x E

ETI{x # 0}

is finite }.

Given a subset A of T , define eA'=eT'T

~IK

t,

~ ~ 1 iftEA ( 0 if t E T \ A

'

eTA is called the characteristic Junction of A on T . If A is the singleton set T } (resp. e {t}). {t}, then we write e T (resp. et) instead of e{t g3 (T) denotes the power set of T and gls(T ) the set of finite subsets of T . Given x E IRT , define

EteT x(t)

-- sup { te~Ax(t) A E ~3I(T)} <_ oc,

adopting the convention that

z(t)

:= o.

tEO

E x a m p l e 1.1.2.2

( 0 ) Let T b e a s e t a n d

g~

:= {x E 1KTI x is bounded}.

f ~ ( T ) is a vector subspace of

II.ll~:f~162

IZ T

and

~IR+,

x,

~suplx(t)I tET

is a norm, called the s u p r e m u m norm. t ~

endowed with this norm is a

Banach space. It is separable iff T is finite. We set f ~ := f~(iN).

It is easy to see that e~ is a vector subspace of 1KT and that I].ll~ is a norm. Let (Xn)nE~ be a Cauchy sequence in t~176 For every e > 0 there is an n~ E IN such that IlXm -- x n l l ~

<

1.1 Normed Spaces

13

for every m, n E IN, with m > n~, n > n~. Hence, given t C T , the sequence

(xn(t))ne~ is a Cauchy sequence (and so a convergent sequence) in ]K. Define x'T

>IK,

tD

> limxn(t). u-+co

Then for all n E IN, n > n~, sup ixn(t) - x(t)l < c. tET

Thus x n - x E t~co(T) and I1~ - ~II~ -< c

for every c > 0 and n C IN, n > n~. Hence x C fco(T) and lim Xn -- X. n--+ oo

gco (T) is thus a Banach space. It is obvious t h a t gco(T) is separable whenever T is finite. Given two distinct subsets A, B of T , we have t h a t i]~ - ~]i-

1.

Hence the distance between any two distinct elements of the set {e A I A E (T)} is 1. If T is infinite, then this set is uncountable and gco(T) is not separable.

Remark.

I The assertion t h a t gco(T) is not separable whenever T is infinite will

be generalized in Corollary 6.3.6.16 b). Example

1.1.2.3

( 0 ) Let T be a set. If T is finite then put

~~

:: c(T)-- c0(T):: ~(T).

I f T is infinite, let ~ denote the filter on T consisting of the cofinite subsets ofT,

i.e. 9= {A C T O T \ A is finite},

and define c ( T ) - = {x e gco(T) ] x ( ~ ) converges},

14

1. Banach Spaces

e~

:= c0(T) := {x e e~(T) I lim x ( ~ ) = 0}.

Then c(T) an co(T) are closed vector subspaces of e~176 and therefore Banach spaces with respect to the induced norm. Given x 9 co(T), if x 7~ O, then the set

{t e T I Ix(t)i = Ilxllt is finite and nonempty, c(T) and co(T) are separable iff T is countable. The norm on co(T) defined by the restriction of the norm on f ~ ( T ) to co(T) is

~o,n~ti,~ e ~ o t ~ by

li-llo we ~t e ~ := ~o := c o ( ~ ) ,

~ := ~ ( ~ ) .

Only the last assertion needs proof. Assume that T is countable. The set of linear combinations of the vectors et (t 9 T) and eT with coefficients in (resp. ~ + i ~ ) is countable and dense in c(T). Hence c(T) and co(T) are separable. Given distinct s, t 9 T ,

Hence co(T) and c(T) are not separable if T is uncountable. E x a m p l e 1.1.2.4

i

( 0 ) Let T be a topological (measurable) space. Define C(T) := {x 9 e~(T) l x is continuous},

(B(T) := {x 9 e~(T) i 9 i~ measurable}). Then C(T) (B(T)) is a closed vector subspace of e~(T) and consequently a Banach space with respect to the induced norm. i E x a m p l e 1.1.2.5

( 0 ) Let T be a set and p a real number, p > 1. Put

f

eP(T) := / z c IKr

, ~ Ix(t)l' < ~ }

O'(T) is a vector subspace of IK T and [].[[p: gP(T) --+ IR+,

x,

defines a norm on gP(T), called the p - n o r m , gP(T) with this norm is a Banaeh space. If T is infinite then Card T is the topological cardinality of gP(T). In particular, gP(T) is separable iff T is countable. We put

e~ := e~(~)

1.1 Normed Spaces

15

By the Minkowski inequality,

(

I~(t) + y ( t ) l ~

<__

I~(t)l"

+

ly(t)l"

tEA

for every x, y E ] K T and A E qJs(T ) . Taking the s u p r e m u m with respect to A on b o t h sides, we may replace A by T in this inequality. It follows t h a t t?P(T) is a vector subspace of

IN T

and t h a t II-IIp is a norm.

Let (xn)~E~ be a Cauchy sequence in t~P(T). Given c > 0 there is an n E E IN with IlZm -- x~ll~ <

for m , n

E IN with m > n ~ , n

> n . . We deduce t h a t for every t E T ,

(x~(t))~E~ is a Cauchy sequence. Define x'T

>IK,

t,

> limx=(t). n - - + (x)

Then 1 p

_
Iz~(t) - ~ ( t ) l ~ tET

and so x~ - x E gP(T) and I 1 ~ - ~11~ <

for every n E IN, n > n~. Hence x E t?P(T) and lim xn = x . n--+ (x)

gP(T) is thus a Banach space. If T is infinite then the set A -

{x E ]K(T) I t e T ~

rex(t), imx(t) E~}

is a dense set of gP(T) and Card A - Card T . We have 1

i_

lies - etll = 2~

1

--

16

1. Banach Spaces

for all distinct s, t C T . Hence if B is a dense set of gP(T) then

B n Vo(e,) # 0 for every t E T , and so Card B > Card T . Remark.

m

The spaces gP(T) (p C [1, c~]) are special cases of the LP-spaces of

integration theory. They are precisely the LP-spaces with respect to counting measure on T .

Proposition 1.1.2.6

( 0 )

Let T be a set and take p,q 6 [1,c~[, with

P
IK (T)

is a vector subspace of

]Z T .

b) IK (T) C gP(T)C eq(T)C co(T) C c(T). c)

IK (T) is dense in both gB(T) and co(T).

d)

The vector subspace of e~(T) generated by {en I A C T} is dense in e~(T) .

e)

Ilxll~ ~ IlXllq _~ Ilxllv for every x e gP(T).

f)

If T is infinite, then the restrictions to IK (T) of the norms [1" lip, I1" IIq, [1" I1~ are pairwise non-equivalent. a), b), c), and d)

are

easy to see.

e) The inequality

I1~11~o < II~llq is trivial. For the second inequality, we may further assume that IIXllq = 1. Then

I~(t)l q _< I~(t)l" for every t C T . Thus

1 = E Ix(t)lq <- E Ix(t)lP' tCT

tET

1.1 Normed Spaces

17

1_ p

Ilxll.- 1 _<

I~(t)l ~ tET

whenever n 9 IN and A is a subset of T containing precisely n elements. Hence the restrictions of the norms I]-]lp, I]-]]q, I].lt~ to IK (T) are not equivalent. I

The next proposition generalizes the preceding examples. P r o p o s i t i o n 1.1.2.7 p 9 [1, oo] U {0}. Let

( 0 ) Let (Et)teT be a family of normed spaces. Take

and q" Fp a)

>IR+,

x,

> II(][x(t)l])teT]lp.

Fp is a vector subspace of 1-I Et and q is a norm on Fp. tcT

b)

If each Et (t 9 T) is complete, then Fp is also complete.

c)

If p r oo, then {x e Fp I {t 9 T I x ( t ) ~ 0} is finite} is a dense vector subspace of Fp.

d)

Fo is a closed subspace of Foo. a) follows from the Examples 1.1.2.2, 1.1.2.3, and 1.1.2.5.

b) Let (Xn)nc~ be a Cauchy sequence in Fp. Given c > 0 there is an rn~ c IN with

q(xm-x.)

< e

for m , n C IN with m_> r n ~ , n _> m~. Then, given t E T , ( x . ( t ) ) . ~ Cauchy sequence in Et . Let x denote the element of I-I Et defined by tET

x(t) "= lim x.(t) n---~ (x)

for every t C T. Then x ~ - x

C Fp and

is a

18

1. Banach Spaces

q(xn - x) < c

for every c > 0 and n > rn~. Hence x C Fp, (xn)ne~ converges to x, and Fp is complete. c) follows from Proposition 1.1.2.6 c). d) follows from Example 1.1.2.3.

Corollary 1.1.2.8

( 0 )

i

Let T be a topological space and E a normed

space. Define C(T, E ) " = {x " T

> E i x is continuous and bounded}

and

il~il = sup i~(t)ii tET

for every x e C(T, E ) . Then C(T, E) is a vector subspace of E T and II.il is a norm. If E is complete, then C(T, E) is complete, Remark.

i

Since C(T) = C(T, IK),

the above example is a generalization of the topological part of Example 1.1.2.4. /

( , )0

D e f i n i t i o n 1.1.2.9

The subset A of a metric space T is called

precompact, if for every c > 0 there is a finite subset B of T with A C [.J U~(t). tEB

A subset A of a topological space is called relatively compact if it is contained in a compact set.

Every compact set of a metric space is precompact and relatively compact. Subsets of precompact (relatively compact) sets are precompact (relatively compact). Every precompact set of a normed space is bounded. [

L e m m a 1.1.2.10 ( 0 ) Let T be a metric space and A , B with A C B . Then the following are equivalent: a)

A is precompact.

be subsets of T

1.1 Normed Spaces

b)

Every sequence in A contains a Cauchy subsequence.

c)

Given any e > O, there is a finite subset C o r B

19

with

A C U U~(t). tEC

a :=> b. Let (tn)ne~ be a sequence in A, and take e > 0. Then there is a subsequence (Sn)ne~ of (tn)ne~ with

d(sm, s . )

< e

for every m, n E IN. Using the diagonal procedure, we can construct a Cauchy subsequence of (tn)nE~b => c. Assume that if C is a finite subset of B , then

A q?_ U U~(t). tEC

Construct a sequence (tn)nE~ in A such t h a t d(t~, t . ) >

C

whenever m, n E IN are distinct. Then (tn)ne~ has no Cauchy subsequence, which contradicts b) c ==~ a is trivial. Lemma

1.1.2.11

I ( 0 ) Let A be a subset of the metric space T . Then the

following are equivalent: a)

A is compact.

b)

A is relatively compact.

c)

Every sequence in A has a subsequence, which converges in T . a => b => c is trival. c => a. Let (tn)nE~ be a sequence in A. There is a sequence ( S n ) n ~ in

A with 1

d(~, t.) < -

n

for every n E IN. By c), ( s , ) , c ~ has a subsequence (sk.),E~ which converges in T . Let

20

1. Banach Spaces

t'-

lim sk~.

n---+ (:XD

m

Then t E A

and lim tkn = t . n---+ (x)

m

Hence A is compact.

I %

I

L e m m a 1.1.2.12 t 0 ) Every relatively compact set of a metric space is precompact. The converse implication holds whenever the metric space is complete. m

Take a relatively compact set A of the metric space T. Then A is compact (Lemma 1.1.2.11 b =~ a). Hence A is precompact. Conversely, let T be a complete metric space and A a precompact set of T. Take a sequence (tn)n~r~ in A. By Lemma 1.1.2.10 a :::> b, (tn)ne~N has a Catchy subsequence (s,~)neIN. Since T is complete, (Sn)ne~ converges in T. Lemma 1.1.2.11 c ~ b shows that A is relatively compact. I P r o p o s i t i o n 1.1.2.13

( 0 ) (Fr6chet, 1907) Take a set T . Take p E [1, c~[.

Let A be a subset of g~(T) (resp. co(T), resp. c ( T ) ) . Then the following are equivalent:

a)

A is relatively compact.

b)

Given c > O, there is a finite subset S o f T

with

1__ p

Iz(t)l ~

<

tcT\S

(resp. sup Iz(t) l < z, resp. tET\S

sup

Ix(~) - x(t)l < c)

s,tET\S

.for every x E A .

a ~ b. A is precompact by Lemma 1.1.2.12, so that there is a finite subset B of t~P(T) (resp. c0(T), resp. c(T)) with A

CU yEB

Take a finite subset S of T such that

U~(y).

1.1 N o r m e d Spaces

21

<- C 2

C

(resp.

sup lY(t)l <

~r\s

C

resp.

2'

sup

l Y ( s ) - y(t)l <

,,~r\s

5)

for every y E B . Take x E A . There is a y E B with x E U S (y). Then

Ix(tll" t

<

Ix(t) - y(t)l ~

+

ly(t)l"

t

t

C

C

<5+~<~. (resp. resp.

sup Ix(t)[ < sup I x ( t ) - y ( t ) l + sup ly(t)l <

tET\S

sup

s,tET\S

-- tET\S

tET\S

C

+

5

C

< c

-2

'

Ix(s) - x(t)l < sup I x ( s ) - y ( s ) l + sET\S

+ sup I x ( t ) - y(t)I + ter\s

sup

,,tET\S

l Y ( s ) - y(t)l < 2e

e

-3 + -3 -- e ) .

b => a. Take ~ > 0 and let S be a finite subset of T with 1

I*(t)l"

<

tET\S

(resp.

sup Ix(t)] <

tET\S

C

resp.

-2 '

for every x E A . Take to E T \ S .

sup

,,teT\S

Ix(s)-

x(t)l <

C

-3

)

There is a finite subset B of A such that

for each x E A there is a y E B with

Ix(t) - y(t) ~ < tES

(resp. sup I x ( t ) - Y(t)l < c, tES

resp.

Cp

3

sup

tESV{to}

C

I x ( t ) - Y(t)l < a )

Take x E A and choose y E B fulfilling the above. Then

0

I

22

I. Banach Spaces

IIx

-

yll~ = ~

tET

= ~

Ix(t)

-

y(t)l p =

Ix(t) - y ( t ) l p +

~

teS

Ix(t) - y ( t ) l p <

teT\S

..

< -5- + <~+

Ix(t)lP

t

+

+

t

ly(t)IP

_<e p

(resp. IIx- Ylloo - sup Ix(t) - y(t)l < c , tET

resp. IIx - yll~ = sup Ix(t) - y(t)l < e ) . tET

Hence

A c U u~(y) yEB

and A is precompact. It now follows from Lemma 1.1.2.12 (and Examples 1.1.2.5 and 1.1.2.3) that A is relatively compact. D e f i n i t i o n 1.1.2.14

( 0 )

(Ascoli, 1883)

I

Let .~ be a set of maps of the

topological space T into the normed space E . .T is called e q u i c o n t i n u o u s at the point t of T if for every e > 0 there is a neighbourhood U of t such that

IIx(~) - x(t)ll < whenever x E ~F and s E U. 3r is called e q u i c o n t i nuous if it is equicontinuous at every point of T .

P r o p o s i t i o n 1.1.2.15

( 0 ) Let T be a compact space, A a dense set of T ,

E a normed space, and 3c an equicontinuous set of maps of T into E . Then the topology on :F of pointwise convergence in A coincides with the topology on .T" of uniform convergence on T .

Take x E ~- and e > 0. For every t E T there is an open neighbourhood Ut of t with c

I l y ( ~ ) - y(t)ll < for every y E 9r and s E Ut. Since T is compact and (Ut)tET is an open covering of T , there is a finite subset B of T with

1.1 Normed Spaces

23

T=Uu~. tCB

Given t E B , choose st E A M Ut. Define

H "=

y E ~ ' l t e B ====>I i x ( s t ) - y(s,)ll

<

g

Then H is a neighbourhood of x in .T" in the topology of pointwise convergence in A. Take y E H and s C T . Then there is a t E B with x C Ut. Thus

II~(s) - y(s)ll < II~(s) - x(t)ll + Ilk(t) - ~ ( ~ d l l + c

+ II~(s,) - y ( s , ) l I +

Ily(s,) - y(t)ll + Ily(t) - y(~)ll < 5 g = c.

Hence

u c {y e : r l s e T ~

IIx(~) - y(~)ll < ~},

so t h a t the topology of .T" of pointwise convergence in A is finer than the topology on ~ of uniform convergence on T . Since the reverse relation is trivial, the two topologies on ~ coincide. Theorem

1.1.2.16

( 0 )

I

(Ascoli, 1883; Arzels 1889) Let T be a compact

space. A set of C(T) is relatively compact iff it is bounded and equicontinuous. First assume t h a t ~ is a relatively compact set of C(T). Then ~" is precompact (Lemma 1.1.2.12) and it is therefore bounded. Choose t c T and c > 0. Then there is a finite subset G of C(T) with

yE6

Furthermore, there is a neighbourhood U of t with c

ly(~)-

y(t)l < 5

whenever y C G and s C U. Take x C ~ and s C U. Then there is a y C G with

I I x - yll < ~. Thus c

c

c

I~(~) - z(t)l _< Ix(~) - y(~)l + ly(~) - y(t)l + ly(t) - ~(t)l < 5 + 5 + 5 - ~'

24

1. Banach Spaces

and so 9v is equicontinuous. Now assume t h a t .~" is bounded and equicontinuous. Take an ultrafilter on ~ . Define x-T

~IK

t,

'

~limy(t). y,~

Choose t E T and r > 0. There is a neighbourhood U of t with ly(~)-

y(t)l <

whenever y E $" and s E U . Then

Ix(~)- ~(t)l _< for every s E U. Hence x is continuous and ~'W {x} is equicontinuous. ~ converges to x in the topology of pointwise convergence. By Proposition 1.1.2.15, converges to x with respect to the norm topology. Hence ~" is relatively compact.

I (Sierpifiski, 1928) Let T be an infinite set. Then

L e m m a 1.1.2.17 ( 0 )

there is a set, 91, of infinite subsets of T of the cardinality of the continuum such that A N B is finite whenever A, B E 91 are distinct. Since T is infinite and ~ is countable, we may assume that ~ C T . Given c~ E IR\(~, let (c~n)nE~ be a sequence in Q converging to c~. The set := {{~, I n E

~}1~

E ~\~}

has the required properties. Definition

1.1.2.1s

I

( 0 ) rh~ ~l~b,~i~) d i , ~ i o ~

ol a w~to~ ~pac~

E is the cardinality of an algebraic basis for E . It does not depend on the particular choice of the basis and is denoted by Dim E . A vector space is called f i n i t e - d i m e n s i o n a l ( i n f i n i t e - d i m e n s i o n a l ) if its algebraic dimension is finite 5nfinite). Proposition

1.1.2.19

Let E be a vector subspace of IK ~ and take x E E\IK (~)

with XeA E E for every A C IN, where 9~ :

~ .

~,

~ ~

~(~)~(~)

Then the dimension of E and the cardinality of E are 2 s~ . In particular, for given p E [1, c~] U {0} both the dimension of gP and the cardinality of gP are 2So

1.1 Normed Spaces

Let R be the dimension of E

25

and let 92 be a set of infinite subsets of

x 1(IK\ { 0 }) with C a r d 92 - 2 r176 such t h a t

A n B

is finite for distinct

A,B

c 92 ( L e m m a 1.1.2.17). T h e n

(XeA)Ae~a is a linearly i n d e p e n d e n t family in E so t h a t Card92 < b~. Since _< C a r d E _< C a r d ]K ~ - (2~~ ~~ - 2 ~~176 = 2 ~~ it follows t h a t = C a r d E = 2 ~~ Lemma

1.1.2.20

I

Let E be a vector space of dimension ~ . I f F, ~ O, then C a r d E = b~2~~

Let A be an algebraic basis for E . Given B E q3s(A ) , identify ]K B canonically with the vector subspace of E generated by B . T h e n

E-

U

IKB"

BEq3s(A)

Since C a r d ]K B - - 2 ~~

for every B c q3s(A ) , C a r d E _< C a r d q3g(A) • 2 ~~ - b~2~~ Now _
2r 1 7 6

so t h a t R2 ~~ <_ C a r d E , C a r d E = b~2~~

I

26

1. 13anach Spaces

Proposition

1.1.2.21

Let p E [1, oo[U {0} and let }t be an infinite cardinal.

Then the dimension of gP(bt) and the cardinality of gP(R) are R ~~ . Take f E R~ and x E t?p. Given t E R, define -1

fx(t) "=

0

if f (t) - 0

x(n)

if f (t) r 0

-1

where -1

n := inf f (t). Then A E gV(R). Since the map It ~q x gP

> gP(R),

(f,x),

> f~

is surjective, Card fv(lq) _< Card(It rq x 0 ~ = R ~~ x 2 ~~ = R~~ (Proposition 1.1.2.19). Given f E R~ , define

f'~

>IK,

1

tl

> E 3n he)'(0

Since the map

~(~)

>IK,

A,

>)--~ nEA

is injective, the map lqiN - - + gP(R),

f ~-----~f

is also injective. Hence ~q}~o_< Card t?P(bt), Card

gP(R)

- R~~

Let R' be the dimension of gP(R). By Proposition 1.1.2.19 2 ~o < ~'

1.1 Normed Spaces

27

so that, by L e m m a 1.1.2.20, R' _< Card gP(R)= R'2 s~ _< R , 2 = R' and R ' = Card gP(R) = R~~ C o r o l l a r y 1.1.2.22

I

Let E be a vector space. Let R > 1 be a cardinal number.

If the dimension of E is R ~~ , then there is a set 7) of pairwise non-equivalent complete norms on E with Card7 ) = 2 ~~ .

Take p C [1, ec]. By Proposition 1.1.2.21, we may identify E algebraically with gP(R). The assertion now follows from Example 1.1.2.5 and Proposition 1.1.2.6 f).

I

Remark. There is no Banach space whose dimension is No (Corollary 1.1.6.17).

C o r o l l a r y 1.1.2.23

Let ~ ~ 0 be an ordinal number such that

sup~ #~

nEIN

(~n)ne~ of ordinal

for every strictly increasing sequence

numbers and such that

for every cardinal number R strictly smaller than Re. Then

Re and the set of equivalence classes of norms on IK (~) has cardinality at least that of the continuum. Now

e'

= U e' rlE(

and C a r d ~1 ( ~ ( ) _ ~(~o ,

Card

gl(Nn)= ~o _<~

for every ~7 E ~ (Proposition 1.1.2.21). Hence

28

1. Banach Spaces

RrNo _< Rr ~ ~o

~ - R~ _

R~

The last assertion follows from the above relation and Corollary 1.1.2.22, m

since the dimension of ]K (~) is R~. Proposition of g~

1.1.2.24

For every infinite cardinal number R, the dimensions

and ]K ~ , and the cardinalities of g~(R) and IK ~ are all 2 ~ .

It is obvious that the cardinalities of g~(R) and IK ~ are 2 ~ . Let ~' (resp. W') be the dimension of g~(tl) (resp. IK ~). By Proposition 1.1.2.19, 2 ~~ < R' and by Lemma 1.1.2.20, 2 ~ = Card g~176 = b~'2~~ < R'2 = R' _< R" _< Card IK ~ = 2 ~ Hence ~'-

C o r o l l a r y 1.1.2.25

R" -

m

2~

For every infinite set T

there is a set P

of pairwise

non-equivalent complete norms on IK T whose cardinality is 2 N~

By Proposition 1.1.2.24 the dimension of

]K T

is

and the claim follows

2 cardT

from Corollary 1.1.2.22 since (2CardT)NO __ 2CardTxNo = 2CardT " E x a m p l e 1.1.2.26

m

( 0 ) Let T be a Hausdorff space and Adb(T) the vector

space of bounded Radon measures on T . Then

is a norm which renders M b ( T )

a Banach space.

Now

(ll~ll- o) ~

(I#I(T)= o ) ~

(l#l = o) ~

( # - o),

I1~ + ~ l l - I~ + ~I(T) ~ (1~1 + I~I)(T) - ~I(T) + I~I(T) -I1~11 + I1~11, ]]oz#]]- ]o~#](T)= [c~[ ] # ] ( T ) = ]ozl]]#]]

I. 1 Normed Spaces

29

for every p, ~ C JMb(T) a n d a E IK, which proves t h a t t h e above m a p is a norm. Let ( # n ) n e ~ be a C a u c h y sequence in t h e n o r m e d space 3rib(T). Given c > 0, t h e r e is a p~ E IN such t h a t

I I # m - #~ll < C for every m , n C IN with m > p~, n _> p~. Let 91 be t h e a - a l g e b r a of Borel sets of T a n d ~ t h e set of c o m p a c t sets of T . T h e n

I#m(A)- pn(A)l = I(/zm - #~)(A)I <~ I~m-

for every r

>

0,A

E 9l, and

m,n

#nl(A) <

c IN with m > p ~ , n

> p~. T h u s

( # n ( A ) ) ~ r ~ is a C a u c h y sequence w h e n e v e r A C 91. Defining p-91

>]K,

A,

r limpn(A), n---+ (x~

it follows t h a t

p ( A U B) -

lim #n(A U 13) n---+ O 0

lim ( # n ( A ) + pn(B)) = #(A) + # ( B ) n---+ O 0

w he ne ve r A, B C 9l are disjoint a n d

I#(A)- # n ( A ) l - l imooI#m(A) - #n(A)l G c for every e > 0, A C 91, a n d n C IN w i t h n > n~. Let (Ak)kc~ be a decreasing sequence in 91 whose intersection is empty. Take

c > 0 and

n C IN with

n _> p~. T h e n

I#(&)l ~ Ip(Ak) - P~(Ak)I § I#n(Ak)l <__c + Ip~(Ak)l for every k E IN, so t h a t

lim sup I#(Ak)l ~ c + lim I ~ ( A k ) l -- c . k--+c~

k---rc~

Since c is a r b i t r a r y , lim # ( A k ) = 0. k---~cx~

T h u s # is a measure. Take A E 91 and c > 0. Take n E IN w i t h n >__p~. Since #n is a R a d o n m e a s u r e on T , there is a K E ~ , w i t h K C A a n d

30

1. Banach Spaces

I#~(L)- #~(A)I < for every L C N , K C L C A . T h u s

I~(L) - ~(A)l <_ I#(L)- ~(L)[ + I#~(L)- #~(A)I + I#~(A)- #(A)I <

for every L E N, K C L C A . Hence # is a R a d o n m e a s u r e on T . Take c > 0 a n d n E ]IN with n > p~. T h e n

I#- ~nl(T) _ 4~, so t h a t # -

#n C Adb(T) and

II~- ~11 ~

4c.

Hence # E 3rib(T) a n d lim #n = #. 71---9"00

T h u s .hdb(T) is a B a n a c h space.

I

1.1 Normed Spaces

31

1.1.3 Minkowski's Theorem Proposition

1.1.3.1

( 0 ) Let p and q be norms on the vector space E .

Then the following are equivalent: a) p and q are equivalent. b) The Cauchy sequences with respect to p coincide with the Cauchy sequences with respect to q. c) p and q generate the same topology on E . a =~ b is trivial. b ==~ c. Take x C E and let (x~)~e~ be a sequence in E converging to x with respect to p. For each n C IN, put Y2n :-- X n ~

Y2n- 1 :-- X.

Then (Yn)~e~ is a Cauchy sequence with respect to p. By b), it is a Cauchy sequence with respect to q as well. Since x is point of adherence of (Y~)~e~ with respect to q, (y~)~e~ converges to x with respect to q. It is now easy to see t h a t p and q generate the same topology on E . c =~ a. Assume t h a t for each n c IN there is an x,~ C E with

p(xn) > nq(x,~). Given n E IN, put 1 y n : = nq'x,~'[ ) x n

Then, given n C IN

P(Yn) ~- p(Xn) > 1, nq(xn) q(v~)

=

q(~)

_

1

Hence lim q(Yn) = 0 n---+oo

and so, by c), lim P(Yn) = O,

n--+oo

which is a contradiction. Hence there is some c~ > 0 with p _< a q . Interchanging the roles of p and q, it follows t h a t p and q are equivalent.

I

32

1. Banach Spaces

Remark.

For a substantial part of the theory of normed spaces (which includes

the first four chapters of this book) only the topology generated by the norm, and not the norm itself is significant. (Thus our interest is not really in normed spaces but, in "normable" spaces). In this "topological case" the norm may be replaced by any other equivalent norm. The theory of locally convex spaces is the best context for this theory. For more precise studies, the "geometrical aspect" is very important and we cannot exchange the norm without damaging the results. This is the case, e.g., with Hilbert spaces and C*-algebras, which are treated in the last four chapters of this book. C o r o l l a r y 1.1.3.2

I"

(

0

\

)

Let p,q be equivalent norms on the vector space

E . I f E is complete with respect to p , then it is also complete with respect to q.

II

D e f i n i t i o n 1.1.3.3

Theorem

1.1.3.4

( 0 )

Given n E IN U {O, o c } , we define

( 0 )

(Mi,~ow~i'~ Theorem, ~Sg~).

Norm~ of

finite-dimensional vector space are equivalent. Given n c IN, let p be an arbitrary norm on IK n and I1" II the Euclidean norm on IKn . Furthermore, let el, e 2 , . . . , en be the standard basis vectors of IK n and put

a :=

p(ek) 2

> O.

k--1

Given x := (xk)ke~. C IK n , we have, by Schwartz's Inequality, that 1

p(x) = p

xkek k=a

<

IXklp(ek) < k=~

1

p(ek) 2

IXk[2 k=l

k=l

It follows that

Ip(x) - p(y)l _< p(z - ~) < ~ll~ - yII for all x, y C IK" (Proposition 1.1.1.3). Hence p is continuous. Define K := {x E IK'~ ] IIx]] = 1}, := inf{p(x) l x e K } .

- ~11~11.

I. 1 Normed Spaces

33

By the Bolzano-Weierstrass Theorem, K is compact. So, by Weierstrass' Theorem, there is an x0 E K such that p(xo) = ~ .

But Xo :/: O. Hence 3 > O. Take x E lKn\{0}. Then 1

IIi ~ ~ 1 1 -

1,

so that 1 ~xEK.

Ilxll

Hence <_ p ( l~z )

-

p(x) ,

i.e.

/~llxll ~ p(x). Thus

911.11 ~p ~ o~11-11, i.e. p and

II-II ~r~ equivalent.

Remark.

The crucial step in the above proof was applying the Bolzano-

It follows that all norms on IKn are equivalent. I

Weierstrass Theorem, which, like Minkowski's theorem, no longer holds when the field considered is not locally compact. For example, the "norms" ......

~

~+,

(~,~),

~v/~ ~ + ~ ,

~

~+,

(~,9),

~ ~+v~/~l

are not equivalent. In fact, if ( a n ) n ~ is a sequence in ~ converging to - v / 2 then lim lan + v/-2 91 i --O, n--+O0

but

lira 4 ~ n - - ~ (X)

+ 1 -- 45 # 0.

34

1. Banach Spaces

C o r o l l a r y 1.1.3.5

( 0 ) Every finite-dimensional normed space is complete.

Take n E IN and let p be a norm on IKn . By Minkowski's Theorem, p and the Euclidean norm on IK n are equivalent. Since ]K n is complete with respect to the Euclidean norm, it follows by Corollary 1.1.3.2 t h a t ]K ~ is also complete with respect to p. C o r o l l a r y 1.1.3.6

m ( 0 ) Let E be a normed space. Every finite-dimensional

vector subspace of E is closed. Let F be a finite-dimensional vector subspace of E . By Corollary 1.1.3.5, F is complete with respect to the induced norm and so it is a closed subset of E.

m

1.1 Normed Spaces

1.1.4 L o c a l l y C o m p a c t D e f i n i t i o n 1.1.4.1

35

Normed Spaces

( 0 ) Let A be a nonempty subset of the normed space

E . Given x E E , we set dA(x) := inf ] i x - YII yEA

dA(X) is called the distance o f x f r o m A .

Proposition

1.1.4.2

( 0 ) Let F be a proper closed vector subspace of the

I1~11- 1

normed space E . Then there is an x E E with dR(x) > -2

Take y E E \ F .

and

1

Since F is closed, dE(y) > 0. There is a z E F with

lly- zll

< 2dR(y).

We set 1

x:=~(y-z). IlY- zll

Then

llxll

= 1. Take a E F . Then

xSince z +

1

a = ~ ( y -

Ily- zll

Ily-

Ix-

z) - a =

1

Ily- zll

( y - (z § Ily- zlla)) 9

zlla belongs to F , we deduce that 1

1

1

all -- lly- zll lly- (z + fly - zlla)ll -> lly - zlldF(y) > -~, dR(x) = i n f i x - - all > aE F

C o r o l l a r y 1.1.4.3

--

1 -2 "

i

( 0 ) Every infinite-dimensional normed space E con-

tains a sequence (Xn)nc~ with

llxnll- 1,

1

I I ~ - ~ n l l >__

for distinct elements m, n E IN.

We construct the sequence recursively. Take n E IN and suppose that the sequence has been constructed up to n -

1. Let F be the vector subspace of

E generated by X l , X 2 , . . . , x n - 1 . Then F is finite-dimensional and so E ~ F .

36

1. Banach Spaces

By Corollary 1.1.3.6, F is closed, so, by Proposition 1.1.4.2, there is an x~ E E

IIx~ll

with

= 1 and

1

d~(x~) > -~ Then

Ix n - x m l l >

1

for every m E IN, m < n, which completes the recursive construction. Remark.

I

The above corollary can be improved (see Exercise 1.3.5).

T h e o r e m 1.1.4.4

( 0 )

(F. Riesz, 1 9 1 8 ) E v e r y locally compact normed

space is finite-dimensional.

Let E be a locally compact normed space. We assume that E is infinitedimensional. By Corollary 1.1.4.3 there is a sequence (xn)ne~ in E such that

[[xnI[-1,

Ixm-xni[_

1

for distinct elements m, n E IN. Since

{x E E I Ilxll = 1} is compact (Corollary 1.1.1.4), the sequence (x,)ne~ contains a convergent subsequence, which is obviously a contradiction. P r o p o s i t i o n 1.1.4.5

I

( 3 ) Let A be a nonempty subset of the normed space

E . Then, given x , y E E , IdA(x) -- dA (Y)I ~ IIx - yll.

We have

dA(x) ~ I I x - zll ~ I I x - Yll + IlY- zll for any z E A , so that

dA(~) ~ IIx - yll + dA(y), dA (X) - dA (y) ~ IIx - yll, IdA(x) - dA(y)I ~ IIx - yll.

I

1.1 Normed Spaces

37

1.1.5 P r o d u c t s of N o r m e d S p a c e s

Proposition 1.1.5.1

( 0 ) Let ( E L ) t e l be a finite family of normed spaces and p be a norm on IK I such that given (c~L),ei, (~L)LeI 9 IR~ with c~L <_ ~

for all ~ 9 I ,

p ((C~L)LeI) < P ((flt.)LEI)

9

Then

L6I

is a norm on I-I E~, which generates the product topology. If each EL (~ 9 I) is LEI

separable (rasp. complete) then l-I EL is also separable (rasp. complete). LCI

It is easy to check that the above map is a norm. It is obviuous that a sequence in I-I EL converges to a point x 9 11 E~ iff the projections of this LEI

LCI

sequence converge to the corresponding projections of x We deduce that the above norm generates the product topology of 1-[ EL. The last assertion is easy LEI

to prove.

I

Definition 1.1.5.2

( 0 ) Let (EL)~cI be a finite family of normed spaces.

The norm

1-I EL LCI

>

IR+,

(x~)Le,'

>

suP lxL I LEI

(Example 1.1.2.5, Proposition 1.1.5.1) is called the supremum norm

of the pro-

duct 11 EL and for every p 9 [1, oo[ the norm LEI

IIE

--+ m ,

>

LEI

is called the p - n o r m

IIx ll LEI

of the product

11 EL (Example 1.1.2.5, Proposition tEI

1.1.5.1). These norms are denoten sometimes by I1"

lb.

Th~ 2-~o~m i~ also

called the Euclidean n o r m of the product 11 EL. Unless otherwise specified, LEI

we take the Euclidean norm on the product of normed space.

All these norms generate the product topology (Proposition 1.1.5.1) and they are therefore equivalent (Proposition 1.1.3.1 c =~ a). When our interest

38

1. Banach Spaces

is restricted to "topological aspects" of the theory (i.e. if we are concentrating on "normable" and not on normed spaces) any of these norms on products will do. But as soon as the "geometric aspects" are important, we have to choose a specific one of these norms. For example, the Euclidean norm will be the appropriate one in the case of Hilbert spaces while the supremum norm will be needed in the case of C*-algebras. P r o p o s i t i o n 1.1.5.3

( 0 )

Let E be a normed space and take (~,~ c IK.

Then the map

ExE

>E,

(x,y),

>c~x+~y

is uniformly continuous.

We have

II(~x~ + ,~yl) -(o~x2 + ~y~)ll = IIc~(Xl

z2) + ,/~(yl

-

-

-

y~)ll

<_ I~l IIx~ - ~11 + 19111y, - y~ I _ < (1~1 ~- + iZl~)~ (ll~ - x~ll ~ +ly, = (Ic~l ~ + I~l~) ~ II(x~, y~)-

- y~tl~) ~ -

(x~, y~)l

for all Xl,X2, yl, y2 E E , which proves the assertion. C o r o l l a r y 1.1.5.4

I

( 0 ) If F is a vector subspace of the normed space E

then F is a vector subspace of E .

Let c~,/3 E IK and let p denote the map ExE

>E,

(x,y),

~ax+C~y. --1

- -

By Proposition 1.1.5.3, ~ is continuous and so ~ (F) is closed. Since

_1

FxFc~(F)

c

~(

T),

it follows that - 1

FxF=FxFc

m

cp(F).

Hence m

c~x + f l y - ~(x, y) C F for every x, y E F , i.e. F is a vector subspace of E .

1.1 Normed Spaces

C o r o l l a r y 1.1.5.5 ( 0 )

39

Let E be a normed space, A a subset of E , and

F the vector subspace of E generated by A . Then F is the smallest closed vector subspace of E constaining A . It is called the closed vector subspace o f E generated by A . If A is countable, then F , endowed with the induced norm, is separable.

By Corollary 1.1.5.4, F is a vector subspace of E and it is clear that it is the smallest closed vector subspace of E containing A. Assume now that A is countable and let B be the set of linear combinations of elements of A with coefficients in Q (resp. ~ + iQ). Then B is countable and m

BcFcB.

Thus m

m

m

BcFcB,

so that F -

m

B , i.e. F is separable.

Example 1.1.5.6

Let E be a normed (Banach) space and c~, fl, 7, 6 scalars

such that

~-

f17 # o.

Then E x E

> IR+,

(z,y) ,

>

sup{ II~x + ~YlI, 117x + @IF}

is a (complete) norm.

The map

ExE

>ExE,

(x, y),

,

(~x + &, 7~ + @)

is linear and bijective and the map

ExE

>Ia+,

(x,y),

is a (complete) norm (Proposition 1.1.5.1).

>

sup{ll*ll, Ilvll} n

40

1. Banach Spaces

1.1.6 Summable Families

Definition 1.1.6.1

( 0 )

An ordered set T is called upward (downward)

directed if for every s, t 9 T , there is an r 9 T such that s<_r,

t<_r

(r<_s,r<_t).

If T is a nonempty set which is directed up (down) then the filter on T generated by the filter base {{seTIs>_t}ltET}

({{seTIs<_t}lteT})

is called the upper (lower) section filter of T . Let I be a set and let ~S(I) be ordered by the inclusion relation. We denote by ~i the upper section filter of ~S(I), i.e. the filter on ~S(I) generated by the filter base

{{J e ~:(I) l J ~ K} I K 9 ~s(I)} .

Definition 1.1.6.2

( 0 )

Let (x~)~c, be a family in the normed space E

If the limit

lim x J,~1 tEJ

exists, then the family (x~)~ei is called s u m m a b l e and the above limit is called

the s u m of the f a m i l y (x~)Lei ; we set

E x ~ ' = l iJ,~j mEx, eel

(in E ) .

tCJ

We adopt the following convention: whenever we write ~ x~ we tacitly assume LEI

that (x~)~cI is summable.

if I-{nc~lp
Xn :Z E XL . LEI

1.1 Normed Spaces

Remark.

41

1) Every finite family is s u m m a b l e and its sum in the above sense is

its usual (algebraic) sum. 2) The assertion X -- ~ X tel

t

is equivalent to the following one: for every e > 0 there is a K E ~ S ( I ) such that

whenever J E ~ i ( I ) , and K C J . Proposition

1.1.6.3

( 0 ) A family sup

(C~L)~EI irt JR+

Lc~t

is summable iff

< (x)

JE~I(I) tEJ

and in this case

E

xt--

tEI

sup

Ea~"

JE~3s (I) t~J

If (OQ)~EI is summable then -- lim 3 JE~I

a, =

~EJ

sup E a~. JEr (I) ~EJ

Conversely, if sup

2.~ aL < O0

JEglI (I) ~EJ

then sup E a , JE~s(I) ~EJ

= limEa~, J,~I ~EJ

i.e. (OQ)tEI is summable.

Remark. By the above result, given families of positive real numbers, the sums defined in Definition 1.1.2.1 and in Definition 1.1.6.2 coincide.

42

1. Banach @aces

P r o p o s i t i o n 1.1.6.4 ( 0 ) Let (E;~)~ei be a finite family of normed spaces and for each Ao E L let P~o be the canonical projection 1-I E~ ~ E~ o. A ,~EL

family (x~)~eI in 11 E~ is summable iff for each A E L the family (p~x~)~ci ~EL

is summable, and in this case px ~

x~ = ~-~p~x~

LEI

LEI

for each A E L. The assertion follows immediately from Proposition 1.1.5.1 and Definition 1.1.6.2. 1 C o r o l l a r y 1.1.6.5 ( 0 ) A family (~,)~e, of complex numbers is summable iff the families (re~,)~ci, (im ~,)Lcl are summable and in this case I

~-'~ c~ - ~-~ re c~ + i ~-~ im a~. tEI

tEI

LEI

P r o p o s i t i o n 1.1.6.6 ( 0 ) Let (xL)~e, be a family in the normed space E . If (x~)~c, is summable, then for every E > 0 there is a J E ~S(I) such that


rl

~ -~

for every K E ~3S(I),J C K . Take K E ~ f ( I \ J ) . finite subsets of I containing J , hence

~EK x~

XL_X eEJUK

Then J and J U K

X -- ~ - ' ~ X t

are

c c <~+~=~.

~EJ

Now assume that E is complete and that the condition described above is fulfilled. Then there is an increasing sequence (Jn)nE~ in ~ / ( I ) such that

1.1 Normed Spaces

LEEKx t

43

II 1 n < --

for every n E IN and for every K E ~ S ( I \ J n ) . For each n E IN we put Yn " = ~

xt.

tE Jn

Then Jp\Jn E ~ l ( I \ J n )

and so

Ily~- y . I I -

xt

tEJp

tEJn

1

<-

n

for every n , p E IN with n < p. Hence (Yn)nE~ is s Cauchy sequence. We put x := lim yn. n--+ (x)

Take e > 0. Take n E IN with 1

C

- < n

C

I l y ~ - xll < - .

2'

2

Then

E

B

Xt -- X

~EK

<

~EK\Jn

tEJn

~

+lly~-~ll < ~+~-~

for every K E ~ : ( I ) with d~ C K . The family (x,),~, is therefore summable and x is its sum.

Remark.

I

Let ~ be the map

V:(z)

>E, J, ~ ~ . LCJ

The condition formulated in the proposition says that ~(~i) is a Cauchy filter, so if E is complete p ( ~ , ) converges and (x~)~E, is summable. C o r o l l a r y 1.1.6.7

( 0 )

Let (x~)~c, be a summable family in the normed

space E . Then for any c > O the set

is finite and the set

is countable.

44

1. Banach Spaces

There is a J

9~ 1 ( I )

with

g

X~

< C

for every K e q3S(I\J ) (Proposition 1.1.6.6). We get

{ ~ Z lllx~ll >_e} c J, i.e.

is a finite set. From

Ilz~ll> _~1} we deduce that

is countable. C o r o l l a r y 1.1.6.8

I Let s be a Banach space. If (xt)~E, is a summable family

in E , then (atxt)tE, is summable in E for every (at)rE, E g ~ Take c > 0. Put := sup ilatl.i tel

There is a J C q3y(I ) such that

4(1 + 5) for every K C q3S(I\J ) (Proposition 1.1.6.6). By Proposition 1.1.1.5,
for every K C ~ i ( I \ J ) ,

so that (aLx~)te, is summable (Proposition 1.1.6.6). I

1.1 Normed Spaces

D e f i n i t i o n 1.1.6.9

( 0 )

45

A family (x~)~e, in a normed space is called

absolutely summable if

IIx ll < LEI

i.e. if ([[x~Ii)~e, is summable in IR (Proposition 1.1.6.3).

Corollary 1.1.6.10 equivalent:

( 0 ) Let E be a normed space. Then the following are

a)

E is complete.

b)

every absolutely summable family in E is summable.

c)

every absolutely summable countable family in E is summable.

If these conditions are fulfilled and (x~)~eI is an absolutely summable family in E then

a ==> b & last assertion. Let c > O. There exists a J E ~J.f(I) such t h a t

~EK

for every K E ~ 3 y ( I \ J ) . We get

for every K E g l S ( I \ J ) . By Proposition 1.1.6.6 the family (x~)~ei is summable. We have

for every J E ~ S ( I ) and therefore

b =:> c is trivial.

46

1. Banach Spaces

c =~ a. Let (xn)ne~ be a Cauchy sequence in E . There is a strictly increasing family (kp)pe~u{o} in IN such that

II~.-

1

~,,II

< 2-;

whenever n ~ IN, n > kp. Then 1

pEIN

pEIN

By c), (xkp - xkp_l)pelN is summable. Given n E IN zk. - Zko + ~ - ~ ( z k , - Zko) p=l

so that (xk.)ne~ converges. It follows that (x.).e~ converges. E is thus complete, m P r o p o s i t i o n 1.1.6.11

( 0 ) Let E be a normed space, (xt)tei,(yt)tci be

summable families in E and c~, ~ C IK. Then (c~x~+ j3yt)~ei is summable and

tel

tel

tel

We have (Proposition 1.1.5.3) that aZxt

+ ~EyL

tel

=alimZxt J,~ l

tel

+ ~lim y'~y~ J,q~l

tEJ

tEJ

= lim ( ~ ~-~ xt +/3 ~-'~ y ~ ) = lim y ~ (c~x~ +/3yt) . J,~l

LEJ

P r o p o s i t i o n 1.1.6.12

tEJ

( 0 ) Let

.

J'TdI tEJ (Xt)te I

be a family in the normed space E

and Io -

{t e l J x, # O}.

Then (x,),et is summable iff (x,)te,o is summable and in this case Z

X t -- ~

tel

Xt .

tEIo

We set :-~3/(I)

> q3:(Io),

J,

> IonJ.

Then P ( ~ t ) - ~Io and ExL=Ext tEJ tE~(J)

for every J E q3l(I ) . The proposition now follows.

m

I. 1 Normed Spaces

47

C o r o l l a r y 1.1.6.13 ( 0 ) Let (x~)~e/, (Y~)~eL be summable families in the normed space E such that I A L = O. Define Ztt :---

f xu

[

if # e l

Yu

if p e L .

Then

E zt'--Ex~+EY~"

#EIUL

tel

AEL

For each A e L and each ~ e I , define xa:=0,

y~:=0.

The corollary now follows from Proposition 1.1.6.12 and Proposition 1.1.6.11.

I Proposition 1.1.6.14

( 0 )

The following assertions are equivalent for

every family (c~)~i in IK :

a)

(oQ)~E I is sumrnable.

b)

(oLt)~e I is absolutely summable.

c) For every injective map ~" IN --+ I the sequence (~'~ c~v(k))ne~ convcrk=l ges.

d)

sup Je~s(I)

In particular, the summable and absolutely summable families coincide in finitedimensional normed spaces.

a =:~ b. Assume first IK = IR. We set

Then (OLt)tEI+ , (Ct~)~EI_ , and (--OQ)~EI are summable. By Corollary 1.1.6.13, (la~i)~ei is summable, hence (a~)~e/ is absolutely summable. Assume now IK = (!?. Then (rea~)~ci , (ima~)~ei are summable (Corollary 1.1.6.5), so by the above considerations they are absolutely summable. We deduce sup

~

la~l _< sup E Ireatl + sup E ]ima,] < oc, JEq3f(I) ,EJ J ~ s (I) ~cJ

48

1. Banach Spaces

hence (c~,),ct is absolutely summable (Proposition 1.1.6.3). b => c is trivial. c ~ d. Assume

JEg~I(I)

~EJ

Then there is an increasing sequence (J~),c~ in

liml~o~l

q3f(I) such that

~

n---~ o o

Let qa : IN --+ I be an injective map for which there is an increasing sequence (Pn)ne~ in IN with

for every n E IN. Then lim

~--~a~(k)

n--+ oo

_-~,~limIZo~I

~,

tE Jn

k=l

which contradicts c). d :::> a. Assume first IK = JR. We set I+ := {t E I I o~, _> 0},

I - := {~ E I J c~, < 0}.

Then sup ? . a , JE~I (I+) ~cJ

< cx~,

sup

).(-c~)

< oo,

JEgls(l- )

(OQ)~EI+ and (-c~)~c/_ are summable (Proposition 1.1.6.3). Hence (oQ)tEI is summable (Proposition 1.1.6.11, Corollary 1.1.6.13).

SO

Assume now IK = r

sup i zeEJoo l

JEq3i(I)

Then

sup Iro Co l

JEq3y(I)

sup I E i m c ~ L [ - s u p limEc~Ll< Jcq3I(I) ~cJ Jcq3f(t) ~cJ

JEq3I(I)

sup I E c ~ ] < o o . JEq3f(I) ~cJ

By the above considerations (re a+)~ct, (ima+)+c/ are summable, hence (a+)+ci is summable (Corollary 1.1.6.5).

Remark. a) Let

(OQ)LEI be a family in IK,

I

1.1 Normed Spaces

f "I

> IK,

~t

49

>a~,

and p be the counting measure on I, i.e. p ' ~ S ( I ) ----+ 1R,

J,

> CardJ.

Then (a~)~ei is summable iff f is #-integrable and in this case Ea~=lfdP.

b) A. Dworetzky and C.A. Rogers proved (1950) that if every summable family in E is absolutely summable, then E is finite-dimensional. Example 1.1.6.15 ( 1 ) ( 3 ) Let T be aset, p 9 [1, oc[, a n d x 9 T. Then (x(t)et)tET is summable in gP(T) (in ~ ( T ) ) iff x 9 tP(T) (x 9 co(T)) and in this case the sum is x. Assume first x 9 fP(T) (x 9 co(T)). Then 1

E teA

x(t)et -- x

-

Ix(t)l . t

p

,

(ll

~(t)~,tea

-

c~

sup tET\A

Ix(t)l)

for every finite subset A of T. Hence (x(t)et)tET is summable in gP(T) (in g~(T) ) and its sum is x. Assume now (x(t)et) summable in gP(T) (in t~~ Let c > 0. Then there is a finite subset A of T such that

V x(t)~ EB

P

for every B 9 ~ s ( T \ A ) (Proposition 1.1.6.6). We get EtEWIx(t)lP -- EteAIx(t)lP + sup {EtEB Iz(t)IPlB C ~ / ( T \ A ) } _ <

teA

teT\A

i.e. x C t~P(T) (x E c0(T)).

I

50

1. Banach Spaces

1.1.6.16 Let E be a Banach space and (En)ne~ be a strictly increasing sequence of closed vector subspaces of E . Then the dimension of E l U En is at least 2 s~

Proposition

nE IN

By Proposition 1.1.4.2, there is a sequence (an)ner~ in E so that

lanlI = 1

an E E n + l ,

1

dEn(an) >

'

2

-

for every n E IN. We set

U " ~cr

~E,

x,

~y~z4(-~an

nEIN

(Corollary 1.1.6.10 a =:v c). u is linear (Proposition 1.1.6.11). Let x E t~~ so that ux E U En . Then there is a p E ] N so that ux E Ep . Let q E ] N , q > p ,

nEIN

such that x(q) # O. Then

4q( q

x(q)

u x - ~-~--~-an

)

E Eq,

n--1

so that 1

- <

,

~dEq< a(q)

aq ~(q)

U X - - n--1

an

-

--

-

4q

I~(q)l

ux- ~

< ix(q )

an II

n=l

n--q+l

ix(q)l 4q

--~an

Ix(q)I < ~ 2 -

IIn=q+l ~ X4(--~an II -<

-ix-(~)i

n--q+l

~

,

Ix(q + n)I 4n

n--1

Let 91 be a set of infinite subsets of IN having the power of continuum such that A M B is finite for every A , B E 91, A # B (Lemma 1.1.2.17) and let F be the vector subspace of e ~162generated by (eA)Ae~. Since (eA)AE~ is linearly independent the dimension of F is 2 s~ . Let y E F \ { 0 } . There is a finite subset if3 of 91 and a family (aB)Sc~ in IK\{0} such that

yChoose A E ~

with

~

O~BeB"

BEff~

1.1 Normed Spaces

[ a A I - sup

BEf~

51

Io~1,

and p E IN with

U (B n c) c IN,,. B,C6~B

Br Given q E A with q > p , we have t h a t E n---1

[Y(q+ n ) [ < 4n --

[OLA[ : n----1

[OLA] <

4~

3

[OgA[ 2

[Y(q)] 2

By the above considerations, uy ~ U E~ and so the m a p nEIN

F

>E/UE.

,

y,

>vuy,

nE ]N

where

>E/UE

vE

nE IN

is the quotient map, is injective. We conclude that the dimension of E l U En nEIN

is at least the dimension of F , i.e. at least 2 ~~ .

Corollary 1.1.6.17

1

The dimension of an infinite-dimensional Banach space

is at least 2 ~~ . By Corollary 1.1.3.6, every infinite-dimensional normed space has a strictly increasing sequence of closed vector subspaces and hence the corollary follows from Proposition 1.1.6.16. Proposition

1.1.6.18

I

Let F be a vector subspace of the normed space E .

If ~ > 1 is the dimension of F , then Card F < R ~~ . By L e m m a 1.1.2.20, Card F - R2 ~~ so t h a t the set of sequences in F has cardinality

(~2~o)~o = ~o2~o _ ~ o . Since every point of F is the limit of a convergent sequence in F , it follows that Card F < R ~~ .

I

52

1. Banach Spaces

P r o p o s i t i o n 1.1.6.19

Let E be an infinite-dimensional normed space of di-

mension N. If there is an increasing sequence (R(~))neN of cardinal numbers with

R = sup R(n) nEIN

such that }to

RCn) < R for every n E IN, then there is a strictly increasing sequence of closed vector subspaces of E whose union is E . In particular, E is not complete and

~ o #: ~. Let (x~)~E}1 be an algebraic basis for E . For each n

E

IN, let En denote the

closed vector subspace of E generated by (x~)~e}1(~). By Proposition 1.1.6.18, }1o the dimension of E~ is at most b~(~), so that En r E . (En)ne~ contains a subsequence with the desired properties. The last assertion follows from the first one, Proposition 1.1.6.16, and Co-

rollary 1.2.2.22.

I

C o r o l l a r y 1.1.6.20

Let ~ be an ordinal number with

1'I2~ < N(+wo 9 Then }1o

N~+n < b~+~o for every n E Wo, }1o

l~+~o

#

R,+~o ,

and there is no Banach space of dimension Rr

We have that

el and for each 7/E

=

U

R(+I, Card

eI (~+1)

=

~l~o f+l

,

1.1 Norrned Spaces

53

Card gl(r/) • (Card 77)r176< b~ ~ < Rr (Proposition 1.1.2.21). Hence lq~+l _

R~+.,o 9

By complete induction b~r n < Rr for each n C w0. The last assertions follow from the above relation and Proposition 1.1.6.19. m

C o r o l l a r y 1.1.6.21

Let E be an infinite-dimensional vector space. I f ~ o < ~+~o

for every ordinal number ~ (this condition follows from the generalized continuum hypothesis), then either there is no complete norm on E or there is a set P of pairwise non-equivalent complete norms on E which is of the power of continuum.

Let ~ be an ordinal number such that the dimension of E is lq~. If ~ = 0 then E admits no complete norm (Corollary 1.1.6.17). If there is an ordinal number r/ such that ~-- ~/+ w0 then, by Proposition 1.1.6.19, E once again admits no complete norm. If does not fulfill either of the above hypotheses, then, by Corollary 1.1.2.23, there is a set P of pairwise non-equivalent complete norms on E which is of the power of continuum. The generalized continuum hypothesis implies that ~ o < 2~ = R~+l < R~+~o for every ordinal number ~.

m

Remark. The assumption in the corollary does not follow from the usual axioms of the set theory, since

2~o < R~o.

54

1. Banach Spaces

D e f i n i t i o n 1.1.6.22

( 0 ) Let E be a Banach space. A power series in E co

is an expression of the form ~ U x n , where t is a variable and (zn)ne~u{0} n--O

is a family in E . If the family (cd~xn)ne~u{0} is absolutely summable for an

aCIK

(0 ~

then we set co E

o{n x n :---

n=O

E

o~n x n

nEINU{O}

(Definition 1.1.6.2). The number

(~

1

:=

lim sup Ilx.ll x

CX:),

1

--

O0

:---

0

)

n.--~ Co

co

is called the radius of convergence of the power series ~ t~xn. n--O oo

T h e o r e m 1.1.6.23

( 0 ) Let ~ t " x . be a power series in the Banach space n--O

E , and let r be its radius of convergence. Then, given ~ E U~(O), the family

(a~X~)neiNu{0} is absolutely summable and the partial sums converge uniformly on ompaa of Urn(O) that (x)

Urn(O)

, E,

.,

' rt--O

is continuous. The family (anx,),~e~u{o} is not summable is a e ]K\U~(O).

Take p C [O,r[ and o~ C Up~(O). Then limsup]]a~x~]] 1 = limsupla[ ]]xn i_1 __ _p < 1. n---}oo

r~---~(:x)

r

By the Cauchy Root Test, the family (cd~xn)nc~u{o} is absolutely summable for every a E Urn(0) and the partial sums converge uniformly on the compact sets of Urn(0) to the sum oo

uy(o)

)E, n--O

Now take a E 1K\Ur~(O) 9Then limsup [a~x~[l! = I~]limsup[]x~] • n - - ~ c<)

r~--+ cx~

la-j[ > 1. r

By Cauchy Root Test, the family (anxn)n~U{0} is not summable.

I

1.1 Normed Spaces

D e f i n i t i o n 1.1.6.24

55

Let E be a Banach space, U a subset of IK, a a n o n -

isolated point of U , and f

9U --+ E .

We say that f

is di~erentiable at

ff hier

1

lim

(f(/7)- f(a))

exists and this limit is called the derivative o f f at a . It is denoted by i f ( a ) . f is called differentiable on U if it is differentiable at every nonisolated point

of U. If f is differentiable at a then it is continuous at a . P r o p o s i t i o n 1.1.6.25

Let ~ tnxn be a power series in the Banach space n--O

E , r its radius of convergence, and oo

<~,

f " Ur~(O) - - + E ,

> ~o<"~,,. n=O

oo

Then the power series ~ nt n-lxn has radius of convergence r , f is differenn=0

tiable, and (DO

n=-O

for every a C U~ (0).

We have that

,im ~u~ (ol,XnJ,)-' - ( aim n~) lim sup Ilxo I1: n--+cx)

n--+oo

rt----~c<)

1

r

(x)

so that the radius of convergence of the power series ~'~ n t n - l x n is r . n=0

Take a e U~(O) and ~ e U ~ ( O ) \ { a } . Then

#

' -

(s/~/-

s/o/) -

o~

' #

X n

n=0

~

- - O/

-

~oxo- ~o:~o '~

,~=o

----

n=0

\m=0

n=o

)

-

56

1. Banach Spaces

E]I~I,~[. There is a p E IN with

(Proposition 1.1.6.11). Let e > 0 and ro O(3

n=p+l

nr'd-lllx~ll

c < 5

Suppose that /5' E Uro_l~l(a). ~ Then ]/31 < r0, and so n-1 E

n-1 ~m(:~n-m-I

n--1

<_ E

m=0

19iml~

"-''-'

<-

m=0

E

1 -- ?2r0n - 1

r 0..... 7"0

m=0

1

n=0

Zmo~--"'-I

(\m=O ~ 2mo~ )-m-1 xo II ~ Z moan-m-' I1-,,11<_

x~ <

n=p+l \m=O

n=p+l m=O oo

n=p+l (Corollary 1.1.6.13, Corollary 1.1.6.10). There is a 5' > 0 with ~ m o~n-m-1

X n --

n=0 \m=O

?.,,ctn - l x n

n=0

<- c 3

for every /3 E U ~ ( a ) . Put 6 "-inf{5', r o - I~1}Then, given fl E U~K(a),

f3

1

n=O II

(f(t3)- f(a))- Z nan-~mn <-

ft..!_ Oz( f ( / 3 ) - f ( o z ) ) - ~n=0 \m=0(~-}~/3rnctn-m-1)Xn -t-

~mozn-m-1

n=0 kin=0 cx~ E

+

Xn -- Z

noLn-lxn

-t-

n=0 g

~-'z, n=p+l

C

C

<5+5+5-e.

Since c is arbitrary O(3

S'(
n<e'-'x,,.

I

1. i Normed Spaces

57

Proposition 1.1.6.26

( 5 ) Let I be an interval in IR and (x~)~, a family of continuously differentiable scalar functions on I . If (X~)LEI and (x'~)~ei are summable families in C(I) then ~ x~ is differentiable and tel

Ex: tel

tel

Take a, b C I. Then E

x~ (b) - E

rE[

~EI

x~ (a) - E (z~ (b) - x~ (a)) = E tel

~EI

Hence ~ x~ is differentiable and tCI

tEI

tEI

x: (t)dt =

E tEI

x: (t) dt.

58

1. Banach Spaces

Exercises E 1.1.1

Prove that all norms on IK are proportional.

E 1.1.2

Show that the map

(U

>IR+,

a,

> Irea I + l i m c ~ I

is a norm when ~ is endowed with its canonical structure as real vector space, but it is not a norm if r E 1.1.3

is taken to be complex vector space.

Let n C I N . Show t h a t there is a norm p on ]K 2 such that

(

) 1

p (1,0) < -I1(1,0)11, n p((0,1)) > nil(o, 1)11. E 1.1.4

Let

E

be the vector space of continuously differentiable maps

] - 1, 1[--, IK with bounded derivatives. Show that E E

>

IR+,

9,

sup ]dx

tCl-l,ll ~(t)

]

> IR+,

te ]- 1,1[ are equivalent norms. E 1.1.5

Show that a normed space E is separable iff it possesses a dense

vector subspace with a countable algebraic basis. E 1.1.6

Show that IK ~ admits no norm which generates the product topo-

logy. (Hint: Given a sequence (an)ne~ in IK, the sequence (ane~)neiN converges to 0 with respect to the product topology.) E 1.1.7

Let T be an infinite set and given f E ]0, oc[ T , define

qs'IK(T)

>IR+,

x,

>E f(t)ix(t)l. tET

Prove the following: a) For each f C ]0, oc[ T , q/ defines a norm. We write IK(T) for the vector space IK (T) endowed with this norm. b) If f, g C ]0, co[ T then:

1.1 Normed Spaces

59

b 1) IK~T) and IK~T) are isometric iff f = g.

b2) qI and qg are equivalent iff inf f (t)

tGT- ~ ~

O,

g(t)

tETinff - ~

~: 0.

c) ]0, c<)[T contains a family ( f ~ ) ~ m such that the norms (qso)oe~ are pairwise inequivalent (Hint: use Lemma 1.1.2.17). d) Given f C ]0, co[ T and p c ]1, c~], the norm qf and the norm induced on IK (T) by t~P(T) are not equivalent (Hint: use Proposition 1.1.2.6 f)). E 1.1.8

Take n E I N

and p E ]0,1[. Show that the map 1

~

~§

(x~)~o,

~

I~1 ~ k=l

isanormiff E 1.1.9

n = 1.

Take p e [1, oc[ and let C([0, 1]) be endowed with the norm 1

C([0,1])

~ IR+ ,

x, ~ ~

(/o

[x(t)[Pdt

Show that C([0, 1]) is separable but not complete. E 1.1.10 Let T be a subset of IR, endowed with the induced topology. Show that C(T) is separable iff T is compact. E 1.1.11

Take x C g~([0, 1]) and consider p : C([0,1])

> IR+,

y,

> sup I x ( t ) y ( t ) l . tE[0,1]

Show that: a) p is a norm iff {z :/- 0} is dense. b) If p is a norm then C([0, 1]) is separable with respect to p. c) If p is a norm then C([0, 1]) is complete with respect to p iff there exists an a > 0 such that {Ix[ > a} is dense. E 1.1.12

Let b~ be a cardinal number such that C a r d { A C R [ C a r d A < R} = R.

Show: a) There exists a set 9.1 of subsets of R such that C a r d 9 . 1 = 2 ~,

CardA=R,

Card(ANB)<

for all A, B C 9/, A r B (generalization of Lemma 1.1.2.17). b) The dimension of e~ and the cardinal number of e~(R) is 2 ~

60

1. Banach @aces

E 1.1.13

Let F be a closed proper vector subspace of the normed space E

and ~ > 0. Show that there is an x E E with IIx[I = 1 and dE(X) > 1-- C.

E 1.1.14 Let (xt)tei be a summable family in the normed space E and the filter of cofinite subsets of I . Show that lim xt = 0. E 1.1.15

Let

(Xt)~)(t,)~)EI•

be a summable family in the Banach space E .

Prove that

E tEI

XtA 9

AEL

(t,~)ElxL

E 1.1.16 Let (xt)~ei be a summable family in the normed space E and f " L --~ I a bijective map. Prove that -- 2___,xt. ~EL

tel

E 1.1.17 Let ( x ~ ) n ~ be a sequence in the Banach space E . Show that the following are equivalent: a) (X,~)n~ is summable. b) The sequence ( ~

XS(k))

k--1

is convergent for every bijective map nEIN

f :IN~IN. Show further that if these conditions are fulfilled, then for every bijective map f : IN--+ IN, E nEIN

xn -- lim ~ x l ( k ) . n---~oo

k-1

1.2 Operators

1.2

61

Operators

In the language of category theory, normed spaces are the objects and operators the morphisms of the category of normed spaces, where an operator is a continuous linear map between normed spaces. It is the operators which are the subject of this section. An i m p o r t a n t and useful feature of the set of operators between two normed spaces is t h a t it too admits a natural normed space structure.

1.2.1

General Results

Proposition

1.2.1.1

( 0 )

(F. Riesz, 1911) Let u " E --+ F

be a linear

map from the normed space E to the n o r m e d space F . Then the following are equivalent:

a) u is continuous. b) u is continuous at O. c) u is continuous at a point of E . d) There is an c~ c lR+ with

f o r every x 6 E . e) u ( E #) is bounded.

f) u is uniformly continuous. f =~ a ==~ b => c and

d r

e are trivial.

c ::v d. Suppose u is continuous at x0 c E . Then there is a 5 > 0 such that

II~x- ~011 _< 1 for every x e E with I I x - ~011 _~ a. We put Ot

"--

1 5

--.

Take x E E\{O} and let 6

Y "-- X o + v~x-~X.

Then

62

1. Banach Spaces

lly - ~011 = and so

mlluxll Ilxll

( )11

=

9

-II~y-~xoll

_< 1,

1

II~xll < ~llx I = ~1 x l l d ::, f. Since

for every x, y E E ,

u is uniformly continuous.

C o r o l l a r y 1.2.1.2 Every infinite-dimensional normed space admits a discontinuous linear form. Let A be an algebraic base of the infinite-dimensional normed space E . Let f : A ~ IK be unbounded and let x' be the linear form on E defined by

x'(x) := f(x)[[x[[ 1 for x E A. Since i-~x c E # whenever x E A,

f(A) C x'(E#). Hence x'(E #) is not bounded and x' is not continuous (Proposition 1.2.1.1 a~e). D e f i n i t i o n 1.2.1.3

( 0 ) Let E and F be normed spaces. We define

L(E, F ) : = {u: E

~ Flu is linear and continuous},

~:(E, F):-

F) I u(E) is finite-dimensional},

{u: s

and

ilull

:-inf{c~ C IR+ J x C E ==, Iluxll < c~llxll }

(the n o r m o$ u)

/or u e L ( E , F ) (Proposition 1.2.1.1 a ::~ d). The elements of s called operators f r o m E to F . We further define

z' := L(E,~:),

c(z):= L(E,E),

C:(E) = C:(E,E),

are

1.2 Operators

1 :=IE:E----+E,

x~

~x

63

(the identity operator on E )

and

(~, x') := (~', ~/:= z'(~) for (x, x') C E x E ' . E' is called the dual of E (H. Hahn, 1927). The elements of E ( E ) are called operators on E . Remark.

a) Because of the equivalence a r

e in Proposition 1.2.1.1, the ope-

rators from E to F are often referred to in the literature as bounded operators. b) It will be proved in Theorem 1.2.1.9 a) that the map >]R+,

E(E,F)

u,

>llull

is a norm. Proposition

ue s

1.2.1.4

( 0 ) Let E and F be normed spaces and take

F).

~) 9 e E ~ II~xll < il~ll IIxil. b) II~ll = sup II~ll = sup Iluxll = s u p , xCUI(O)

xEE#

xEE

Ilxll=l where in the last equality it is assumed that E # {0}. c) If G is a subspace of E , then li~ l all _< II~li.

a) We put A := {~ c IR+ I x E E ==~ lluxll <_ ~llxll}. Then

II~xll < --

inf

aEA

~llxll- Ilult I1~11

for every x C E . b) We put xEU1 (0)

Take x E E and /3 >

lixll.

Then 1

~x

1

= p-~llxll . . . . < 1,

64

1. Banach Spaces

so that we deduce successively

fill~xll-

u(z)

_<~,

Iluxll <~ ~,

Iluxll ~< ~llxll,

Ilull ___ ~,

since /3 and x are arbitrary. Now take x 9 E # . By a),

II~xll < I1~1 Ilxll < Ilull. Thus

I1~1 ~< ~ ~< sup Iluxll ~< Ilull xEE#

and

I1~ I =

sup

xEU1 (0)

I1~11 = sup II~xll. xEE#

Now assume that E # {0}. Take x e E # \ { 0 } . Then 1

II~xll < ii--~llII~xll =

(1)11 ii-~ll 9

< sup yEE II~vll. Ilyll=l

so that

sup II~xll ~ sup.

xEE#

xEE

II~xll The reverse inequality is trivial. c) We have, by b), that

I1~ I GII = sup II~xll ~ sup II~xll- II~ll, xcG#

C o r o l l a r y 1.2.1.5 u e s

F),v

9s

( 0 ) Let E , F ,

xcE#

and G be normed spaces and take

G). Then v o u 9 s

G) and

I1~ o ~11 _< II~II II~ll. v o u is linear and

II~ o ~(~)11-

I~(~)ll _~ Ilvl II~ll _~ IIv I I1~11 II~ll

for every x C E (Proposition 1.2.1.4 a) ). The corollary now follows from Proposition 1.2.1.1 d =~ a. I

1.2 Operators

/

C o r o l l a r y 1.2.1.6

65

\

( 0 ) Let E be a normed space and take x' E E ' . Then

re o x' is a continuous ]R-linear form on E and

1Ire o x'li-

llx'll 9

re o x is obviously continuous. We have that re o x ' ( a x +/3y) = r e ( x ' ( a x +/3y)) = re(c~x'(x) + ~x'(y)) = = a re(x'(x)) + ~ re(x'(y)) - c~reo x'(x) + / 3 r e o x'(y) for x, y C E and a,/3 E JR, i.e. re o x' is ]R-linear. It follows from Proposition 1.2.1.4 b) that [Ire o x ' l J -

sup Ire ox'(x)l < sup [x'(x)[ =/Ix'll. xE EC/:

xE E #

By Proposition 1.2.1.4 a), Ix' (x)i ~ = x' (x)x' (x) = x' (x' (x)x) - re(x' (x' (x)x)) = ~

o x'(.'(x).)

<

I[~e o *'ll II*'(*)~[I - I1~ ~ *'ll Ix'(*)l Ilxll,

so that ]x'(x)] < [Ire o x'l] I]x ] for x r E . It follows that

iix'i] _< iir~ o ~']] and

ilre o x'li = II*'il 9 Proposition

1.2.1.7

set, ~ a filter on T ,

( 0 )

Let E

m

and F be normed spaces. Let T be a

(ut)t~T a family in s

such that

lim utx t,~

exists for every x C E , and

u'E

>F,

x~

>limutx.

u is linear and if

lim inf ilu, li < o~, then u is continuous and

[u[[ _< lim inf [lut[I 9 t,;~

66

1. Banach @aces

We have that u(o~x + ~y) = limt,;~ut(o~x + j3y) = l~,~(oLutx + j3uty)) = ozux + ~uy for x, y C E and a,/~ C IK (Proposition 1.1.5.3), i.e. u is linear. Moreover

Iluxll : 1~,~ II~xll ~ lim lmf II~[I IIxll - IIxll limte;~inf IiutI[ for every x E E (Corollary 1.1.1.4, Proposition 1.2.1.4 a)), which proves the last assertion.

I

Lemma 1.2.1.8

( 0 )

Let E and F be vector spaces. For every x C E ,

the map FE

>F ,

u:

> ux

is linear (it is called the e v a l u a t i o n o f x ) and {u E FE l u is linear} is a vector subspace of F E . The first assertion is a consequence of the definition of the vector space structure on F E . Let u , v be linear maps of E into F and put w:--uWv. Then w ( a x +/~y) = u ( a x + fly) + v ( a x +/3y) = a u x + 3 u y + a v x +/~vy = = ~ ( ~ x + v~) + ~(~y + vy) = ~ x

+ ~wy

for x , y E E and a, 3 c IK, i.e. w is linear. Let u" E --~ F be linear. Take 7 C ]K, and put v "= 7 u . Then

~(~x + Z~) - 7 ~ ( ~ x + / ~ )

- 7(~

+ 9~)

- ~

+ 9vy

for x , y C E and a, fl C ]K, i.e. v is linear. It follows from the above considerations that {u E F E ] u is linear} is a vector subspace of F E .

I

1.2 Operators

Theorem a) s

1.2.1.9

67

( 0 ) Let E and F be normed spaces.

F) is a vector subspace of F E and the map s

>JR+,

u,

>llull

is a norm; it is called the canonical n o r m oJ' /:(E, F ) .

b) If F is complete, then s a) Take u, v C s

II(u + v)~ll-

F) is also complete.

F ) . Then

Ilux + vxll ~ I1~11 + Ilvxll ~ Ilull Ilxll + Ilvll Ilxll = = (11~[I + Ilvll)llxll

for every x C E (Proposition 1.2.1.4 a)), so that u + v c /:(E, F ) (Lemma 1.2.1.8, Proposition 1.2.1.1 d =~ a) and

II~ + ~ll -< II-ll + Jl~Jl. Take c~ E IK. Then

sup I(~u)xl-

I~1 sup Iluxll = I~l Ilull

xEE#

xEE#

(Proposition 1.2.1.4 b)), so that c~u E F_.(E,F) (Proposition 1.2.1.1 d =:> a, Lemma 1.2.1.8) and

II~tl-

I~l Ilull

(Proposition 1.2.1.4 b)). Finally, it is easy to see that u - 0 whenever

Iluil

= o.

b) Let (Un)ne~ be a Cauchy sequence in E ( E , F ) . For every c > 0 there is an m~ such that Ilun-upl I <e for every n , p E IN, with n , p > m~. Take x E E . Then

for every c > 0 and each n , p C IN with n , p >_ m~ (Proposition 1.2.1.4 a)). Hence

(UnX)nelN is

a Cauchy sequence and therefore a convergent sequence in

F . Define u'E

.....> F ,

x,

> limunx. n -----~(:x3

68

1. Banach Spaces

Then _(u~ - u)x. = u n x -

ux = u ~ x -

limupx p--~ o o

= lim (unx - UpX) p-+ oo

lim (un - Up)X p----> c ~

whenever n E IN and x E E . By Proposition 1.2.1.7, un - u E s

F) and

II~.- ~ll < for c > 0 and n E IN, n > m~. It follows t h a t u E s

F) and

lim Un - u .

n---> ~

s

I

F ) is thus complete.

C o r o l l a r y 1.2.1.10

( 0 )

The dual space of a normed space is a Banach

I

space.

C o r o l l a r y 1.2.1.11 D e f i n i t i o n 1.2.1.12

( 0 ) ( 0 )

If E is a Banach space, then so is f~(E). Let E , F

I

be n o r m e d s p a c e s and u " E - + F

a

bijective linear map. u is called an i s o m e t r y o f n o r m e d spaces if

II~xll- IIxll for ever'9 x E E . u is called an i s o m o r p h i s m o f n o r m e d spaces if u and u -~ are continuous. E and F are called i s o m e t r i c ( i s o m o r p h i c ) if there exists an isometry (isomorphism) E - + F .

If u : E --+ F is an isometry (isomorphism), then so is u -1 . Every isometry is an isomorphism and every isomorphism is a homeomorphism. Isometry is equivalence in the category of normed spaces (geometrical aspect), whereas isomorphism is the equivalence relation in the category of normable spaces (topological aspect). We may replace "bijective" by "surjective" in the definition of isometry, since the main property of isometry implies that the injectivity follows. Let T be a finite set with at least two elements and take p,q E [1, oo] with p ~= q. Then gP(T) and gq(T) are isomorphic (Minkowski Theorem), but they are not isometric. This can be seen by considering their unit balls. If T is infinite then ~P(T) and ~q(T) are not even isomorphic. There exists an infinite-dimensional Banach space E not isomorphic to any of its proper subspaces (W.T. Gowers, Bull. London Math. Soc. 26 (1994), 523530). Then E is not isomorphic to E • ]K.

1.2 Operators

Proposition

1.2.1.13

1

\

0

(

69

) Let F be a dense subspace of the normed space

E and G a Banach space. Then the map

~(E, a)

>~(F, a) ,

> uIF

u,

is an isometry. It is obvious that the above map is linear. By Proposition 1.2.1.4 b), IlulYll =

for every u C s

sup

x6F#

Iluxll :

sup

x6E#

II~xll :

I1~11

since F # is dense in E # . Take v C s

By

Proposition 1.2.1.1 a :=v f, v is uniformly continuous. Since G is complete, there exists a continuous extension u : E --+ G of v. u is then linear, i.e. u E s

and uiF = v. Hence the m a p

~(E, G)

>~(F, G) ,

> ulF

u,

I

is surjective and so it is an isometry. Proposition

1.2.1.14

Let E, F be isomorphic normed spaces. If E is com-

plete then so is F . Let u : E -+ F be an isomorphism and let (Yn)ne~ be a Cauchy sequence f

%

in F . Then ( u - l y n ) \

/ nCIN

is a Cauchy sequence (Proposition 1.2.1.1 a ==v f ) a n d

so a convergent sequence in E . Hence u ( n--+cx)limu - l y n ) - n~lim U(u-lyn)

- - n---~oclimYn.

I

Thus (Yn)n~lN converges and F is complete. Proposition

1.2.1.15

Let p and q be norms on the vector space E . Let

Ep (resp. Eq ) be the vector space E endowed with the norm p (resp. q ), and define u:Ep--+ Then the following are equivalent: a)

u is an isomorphism.

b)

p and q are equivalent.

Eq,

x~

>x .

70

1. Banach Spaces

a :=~ b. Given x c E ,

q(x) = q(ux) < Ilullp(x), p(x) = p(u-~x) < Ilu-~llq(x) (Proposition 1.2.1.4 a) ), i.e. p and q are equivalent. b ==~ a. For some a > 0 1

- p < q <_ c~p. Thus, given x C E

q(ux) = q(x) < ~p(x), p(u-lx) = p(x) < (~q(x) . Therefore b o t h u and u -1 are continuous (Proposition 1.2.1.1 d =~ a).

I

1.2.1.16 ( 0 ) Let E and F be normed spaces and (x~)L~, a summable family in E . Take u C f_,(E, F). Then

Proposition

We have

U(~XL)

Corollary

-- u ( l i m ~ xCJ L)

1.2.1.17

limu(~xL)

-- J,~ lim~j

I

( 0 ) Let E and F be Banach spaces. Take u e s

F).

(x)

Let ~ tnxn be a power series in E , and r, r' be the radii of convergence of n--O oo E tnxn n=O

oo

and ~

tn~txn,

respectively. Then r < r' and

n=O

?2

oLnxn n--O

for every ~ E U~(O).

-- E oln~txn n--O

1.2 Operators

71

Given n c IN,

il~z~il _< ll~ll IZnJl (Proposition 1.2.1.4 a) ), so that _ - 1_ lim sup ilux~lJ -~ _< lim sup liuli• ]lx~ll I _< lim sup lix~ll I

-,1

r !

r

n--+oo

rt---~(x)

n---~ o o

r
The equality now follows from Proposition 1.2.1.16. Proposition

m

( 0 ) Let E and F be normed spaces, u" E --+ F

1.2.1.18

a linear map, and a a strictly positive real number such that

:for every x 6 E . (This property of u is sometimes called lower bounded.) a)

u is injective.

b)

If u is surjective, then

c)

If E is complete and u is continuous, then u ( E ) is closed.

d)

If E is complete, u is continuous, and u ( E ) isomorphis',n of normed spaces.

?.t - 1

1 is continuous and iJu-lJJ < ~.

is dense then u is an

a) is trivial. b) Take y E F and let x " - u - l y .

Then

li~-~yii: il~ll<_ !ll~xtl : !tl~il (2 s and the assertion follows (Proposition 1.2.1.1 d ==~ a). c) Take y e u ( E ) . There is a sequence (xn)~c~ in E with lim ux~ - y. rt--+ OO

Take c > 0. There is an m C IN such that

lluxm- UXnll <

~

for every n C IN with n > m . Then

IIXm

Xnll < I llU(Xm O~

xn)ll

l ll~

?-tXnll

OL

OLC OL

for every n C IN with n > m . Hence (xn)ne~ is a Cauchy sequence and so a convergent sequence in E . It follows that y-

lim uxn = u( lim xn) e u ( E ) . n---+ OO

n--+OC~

Thus u(E) is closed. d) follows from b) and c).

m

72

1. Banach Spaces

Proposition

1.2.1.19 ( 0 )

For each x' = (x't)tE1

Let (Ee)ee, be a finite family of normed spaces.

YI E~, defi~2e

E

eEI

-" 9 11 E~ ---, ~,

(x~)~,. , Z ~:(x~).

eel

tel 1

~) x~'E (11,~,E,)' a~d IIx~ll--II~'l : ( ~ II~:ll~)~

for every x' C [I E~. eEI

b)

The map ,.,.,

II,:

X I t,

,

eel

) Xt

eCI

is an isometry. a) Given x -

(xe)ee, E 1-I Ee, e6I

I~(x)l- I ~--~~:(x~) I <_ ~ I~:(x~)l ~ ~-~ I ~:ll IIx~ll tEI

eEI 1

_< (}-~ IIx:ll~)~ (~ eel

tEI 1

IIx~l ~)~ -IIx'll I ~ll

tel

(Proposition 1.2.1.4 a ) ) , so t h a t xD 6 ( . ~ , E e ) '

and

I~l < I x'll

(Proposition

1.2.1.1 d => a). Take e > 0. For each t E I there is an xe C Ee such t h a t

I~ll_lx:l, (Proposition 1.2.1.4 b) ). P u t x -

x:(~)>ll~:l~-~ (x~)eeI 9T h e n

IIx~l ~ ~ II~-'ll II~ll ~ Ix'(~)l = = ~

~:(~,)

~ ~

e6I

11~TII~ - ~

Card/-I1~'11 ~ -

~

Card/

eEI

(Proposition 1.2.1.4 a) ). Since c is arbitrary,

II~' ___ x'll, b) Take y' E

Ee

IIx'll-IIx'll.

For each t E I

let ue " Ee ~

llE~

denote the

,kEL

canonical imbedding and put x eI -

yl

o ue,

X ! " - - ( X tt) t E I .

Then

X!

E 11 E~ and eCI

1.2 Operators

~(x) 9 = Z x:(x~)= ~ y'~x~ = ~'(x) LCI

LEI

for every x "-- (x~)~ei e I-I EL. Hence x ' = y' and the map LEI

I-[<~

(n)' .~ .

is surjective. By a) it is an isometry.

x'.

.x,

73

74

1. Banach Spaces

1.2.2 S t a n d a r d E x a m p l e s

Let p E [1, c~] U (0}. The conjugate exponent of p is the number q E [1, oe] defined as follows:

D e f i n i t i o n 1.2.2.1

( 0 )

q := c<)

ifp=l,

q'- ~

if p E ] l , c ~ [ ,

q:=l

if p E {O, oc}.

Two numbers p,q are called conjugate exponents (weakly conjugate exponents) if p,q E [1, co], (p,q E [1, c~]U{0}) and p is the conjugate exponent of q or q ist the conjugate exponent of p. If p, q are conjugate exponents, then p is the conjugate exponent of q and q is the conjugate exponent of p. Proposition

1.2.2.2

( 0 )

Let T

be a

set. Let

p, q

be conjugate exponents.

Take x E IKT and a E IR+. If [E

x(t)y(t) I ~- aIlYlIP

lET

for every y E IK (T) , then x E gq(T) and IIxll~ < ~ .

First assume that p ~ 1. Put

y. T--+ IK,

t

~ x(t)ix(t)lq-2

if x(t) ~ 0

L0

if

x ( t ) - O.

If p = cx~, then q = 1 and for every A E ~3s(T ), E

Iz(t)l = E

tEA

z(t)y(t)en(t) <_ c~IlyeAil~ _< c~

tET

so that x E gl(T) and

If p ~= c~, then for every A E ~ s ( T ) ,

1.2 Operators

E

[x(t)]q = E

tEA

x(t)y(t)eA(t) <_ a]lyeA ]p =

tET

---OL(E IY(t)]P)1 = OZ.(E Ix(t)iP(q-l))~ -t6A t6A tEA

and so 1

t6A

1

1

tCA

Thus x E fq(T) and

Ilxll~ _~ ~. Now take p = 1. Then

Ix(s)l- ~

x(t)eT(t)

<

ozlle~Ill

-

a

tET

for every s E T , so that x E g~(T) and

Ilxll~ _< ~ E x a m p l e 1.2.2.3 a)

( 0 ) Let T be a set and p,q conjugate exponents.

xy 6 6I (T) and

Ilxvll~ __ IIxll~llvll, for x E (q(T) and y E gP(T), where xy" T

> IK,

t,

> x(t)y(t) ;

we define

tET

for every x E ~q(T). b)

~ E gP(T)' and ] ] ~ ] : ]]X]]q for every x E (q(T).

75

76

d)

1. Banach Spaces

If p r oc then the map eq(T)

>ep(T)',

x,

~

is an isometry. e)

The map

el(T) -----+ co(T)',

x,

>~lco(T)

is an isometry.

f)

If ~ is a free ultrafilter on T one-point set), then the map x " e~176

(i.e. an ultrafilter on T possessing no

(resp. c(T)) ---+ IK,

belongs to e~ ', (resp. c(T)') and x ' r every x E t I (T) .

g)

y,

)lim y(~)

~ (resp. x ' r

~ I c(T)) for

The map e ~(r)

~e ~ ( r ) '

(resp. gI (T)

~ c(T)' ,

~,

~

x,

~ ~ I c(T) )

is an isometry iff T is finite.

a) By the HSlder inequality, Z

Ix(t)y(t)[ <- Ilxllqlly[Ip

tcA

for every A C g3s(T ) . Thus E

Ix(t)y(t)[ <- IIx Iqilyl]v.

tET

Hence xy C f l ( T ) and

b) By a), Proposition 1.1.6.11, and Proposition 1.2.1.1 d ~ a, ~ C t~P(T)' and

I1~11 < Ilxllq.

1.2 Operators

77

We have that

]E x(t)y(t)]

-- ~(y)l _< I1~11Ilyll,

tET

for every y

(Proposition 1.2.1.4 a)). By Proposition 1.2.2.2,

E ]I4[(T)

IIxll~ <__ II~ll. Hence II~ll -

llXllq.

c) We have that

I E x ( t ) y ( t ) I = I~(y)] _~ I]xlco(T) iilyll~ tET

for every y E IK (T) (Proposition 1.2.1.4 a) ). By Proposition 1.2.2.2

I1~11, ~ I1~ I ~o(T)ll. Hence by b) and Proposition 1.2.1.4 c)

II~ll- Ilxl}, < II~ I co(T)ll < I1~11, I1~ I ~o(T)ll- I1~11- I1=11,d) (resp. e)). Let x' E t~P(T)', (resp.

x.T

~,

x' E co(T)'). Define t,

>~'(~T).

Then

tET

tET

(resp.

<

I='11 Ily Ioo)

for every y E IK (T) (Proposition 1.2.1.4 a)). By Proposition 1.2.2.2, (resp.

x E gq(T)

x E gl(T)). We have that ~(~T) - x(t)

for every t E T, so that ~ -

x'

on

- x'(~T)

] Z (T) .

Since

] Z (T)

co(T) (Proposition 1.1.2.6 c)), ~ = x'. Hence the map

is dense in t~P(T) and

78

1. Banach @aces

eq(T)

> gP(T)',

x,

~~

(resp. el(T)

~ co(T)',

x,

~ ~[c0(T))

is surjective. By b) (resp. c)) it is an isometry. f) It is obvious that x' E g~ x'-

x'(eT) -- 1, and x ' = 0 on IK (T) . Hence

0 on co(T) (Proposition 1.1.2.6 c)). Assume that there is an x E tl(T)

with ~ = x'. Then ~ = 0 on co(T) and so, by c), Ilxlll - II~ l co(T)ll - O,

x=O.

This leads to the contradiction l=x'(eT)=~(eT)=O.

g) follows from e) and f). Remark.

ll

A representation of t ~ ( T ) ' is given in Corollary 1.2.2.14.

E x a m p l e 1.2.2.4

Let T be an infinite set and ~ the filter of cofinite subsets

o f T (i.~. 9= { A C T I T \ A

x " c(T)

~ IK,

x,

is f i n i t e } ) ,

~ limx(~),

to C T . Let ~ " T \ {to} ~ T be a bqection. We set

_

x "T

~IK

'

t~

f x'(x)

if t - t o

~ ~(~(t)) - ~'(~)

if t # to

for every x C c ( T ) .

a)

x' C c ( T ) ' ,

b)

~ E co(T) for every x C c(T) ; define

IIx'll - 1.

c(T) ~ c o ( T ) , c)

~,

u is an isomorphism and

JJ~ I = IJ~-~ll = 2. d)

co(T) and c(T) are not isometric.

~~

1.2 Operators

79

a) and b) are obvious. c) It is clear t h a t u is linear and injective. We have t h a t

I~(to)l = Ix'(x)l ~ Ilxll~, sup I~(t)l-

sup

tET\ { to }

Ix(~(t))-x'(x)l _~ Ix'(x)l +

tET\ { to }

sup

Ix(~(t))l ~ 211xll~,

tET\ {to }

so t h a t

I1~11~ ~ 211xllo~ for every x 9 c ( T ) . Hence u is continuous (Proposition 1.2.1.1 d =~ a ) and

I1~11 ~ 2. From e T - 2eto 9 c ( T ) ,

IleT- 2etoll~ --- 1,

Ilu(eT- 2~,o)I~ = 2,

we see t h a t ]lull = 2. Let x C co(T). Define y "-- X ( t o ) e T nt- X o (t9-1 e c ( T ) .

Then x'(v)

x(to),

-

so t h a t y--x.

Hence u is surjective and U-I(x)

-- X ( t o ) e T -t- X 0 ~ - 1

We deduce t h a t

I1~-~11~ _< 211xll~ for every x C co(T). Thus

Ilu-lll

U -1

is continuous (Proposition 1.2.1.1 d =~ a ) and

_< 2. From eto + e~-l(to) E co(T),

I1~o + %-1(to)11~ --- 1, it follows t h a t Iiu-ll] = 2. d) Assume t h a t

I1~-1(~o + ~-,(~0))1 ~ -

2

80

1. Banach Spaces

v "c(T) -----+ co(T) is an isometry. Put

A "- { IVeTI - 1} and

At "- {Iv(eT + et)l = 2} for every t C T . Since v is an isometry, A and At are finite, nonempty, and

At C A for every t 9 T (Example 1.1.2.3). Hence there are distinct t', t" 9 T with At, N At,, :/: O. Take t 9 At, N At,,. Then

(vet,)(t) = (VeT)(t) = (vet,,)(t), so t h a t

2-

I(vet, + vet,,)(t)l ~ Iv(et,-+ et,,)Ioo = I1~,' +

e,,,ll~

- 1,

which is a contradiction. Hence co(T) and c(T) are not isometric,

Remark. Example

d) will be generalized in Corollary 1.2.7.18.

1.2.2.5

( 0 ) (F. Riesz, 1910)

Let # be a positive measure and

p, q conjugate exponents. a)

xy 9 L l(#) and

Ilxylll ~ Ilxllqllyltp

(I-I~tder

inequality)

for every x 9 Lq(p) and y 9 LP(p). For each x 9 Lq(#), put ~" LP(#)

b)

m

~ 9 LP(#) ' and [[~[] =

c) ~f p ell, ~[

Ilxllq for

) IK,

y,

) f xyd#. J

~v~y x e Lq(#).

th~n th~ map

L q(p) is an isometry.

> L p(p)',

x,

~

1.2 Operators

d)

81

The map

>LI(#) ',

L~(#)

x~

>~

is an isometry iff L ~ ( # ) is order complete.

The proofs can be found in most books on integration theory.

I

If T is a set and p counting measure on T, then Example 1.2.2.3 a),b),d) becomes a special case of the above example. Remark.

Definition 1.2.2.6

( 0 )

5(s, t) "- (~st "=

We set

S 1 if s = t

I

(the K r o n e c k e r ' s symbol)

0 if s # t

for all s, t .

E x a m p l e 1.2.2.7

( 0 ) Let [Olij]iEINm,jEINn be a matrix over IK. Let

u 9IK n --+ IK TM be the associated linear map and A the set of the roots of the equation

=0. If we endow IK TM and IK n with the Euclidean norm, then:

a) A C JR+. b)

Given A C A and a solution x := ( x j ) j c ~ tions

j=l

i=1

we have that II~xll 2 = Allxll 2 m

c)

I/nil2 - sup A _< E ~ [aijl 2. )~cA

i=1 j = l

of the system of linear equa-

82

d)

1. Banach Spaces

If n = 2 , then

ilull2

1

12

m i=1

-4

i=1

]OZ/1]2 -]- E ]OL/2 2 i=1

i=1

-~-

m

]c~i1[ 2 i=1

[~

-

i=1

-2

--"

E ~ i=1

m

E (J~,~l~ + J~,,J') +

1

i=1

-1-

(IOdil

--1-IOLi2I2)

i=1

O~i20Ljl

-- 2 E [a/10/j2i,j=l

a & b. We have that //1

r

j,k=l i=1

k=l

c) Take x = ( x k ) k e ~ such that Iluzll ~- - s u p { l l u y l l 2 1 y By Lagrange's Theorem, there is a A E IR such that

c II( n, Ilyll = 1}.

Oll~yll ~ = ~ o lyll 2 0~k 0~k for every k E INn when y = z . Thus

j=l

i=1

for every k C INn. By a), b), and Proposition 1.2.1.4 b), ]]u]]2 = supA. ,~EA

Take ,~ ~ A with ) ~ - I1~112 and for every k C INn,

9 c IK ~ such that ( , ) i s

fulfilled. Then

1.2 Operators

83

m

E

aij~ik

j=l i=1

j=l

i=1

-< i=1

Ixjl

i=1

I~,~l ~

~

j=l i=l

I~jl ~

Ilxll.

It follows

11~l1411xll~ =

k=l

II~ll~lx~l ~ _<

)

~ I~1 ~ k=l i=1

m

n

i=l

j=l

~ I~,~1~ Ilxl ~, j=l i=1

d) We have that

m E I~kll 2 -- ~ k=l m

E

k=l

~k2~kl

m E ~kl~k2 k=l m E

k=l

I~k2l ~

-

2

m

_~/~2 (~'~k=lIOLkll2-t- k=l ~ IOZk2'2) )~2r" (~k=l IOZkl]2)(~-~k=llXk212) E IOQIOLj2 -- OQ2OLjl i,j=l

-- E i,j=l

ailOgj2 -- Cti2Ctjl

)(

k--1

-~kl C~k2

ailO~j2 -- O~i20Ljl

)

=

m --i,3.~=1 (OZill 20Lj212-I- 1~i2121OLjll2) -- ~ ('~il~i20LjlOL-j2-~OLilOli2OLjlOlj2)i,j--1 m -~k l Ctk2 : 2 (k~ 1 IOZkl,2/ (k_~l ,OLk212) -- 2 k--1

m

The assertion now follows from c). E x a m p l e 1.2.2.8 ( 0 ) u E/:(IK 2) and let

Take IK 2 endowed with the Euclidean norm. Take

5

be the matrix associated to u.

84

1. Banach Spaces

a) I1~12 =

= 1- ('c~12 + 1/312+ 1'712+ 15[2+ i('oz'2 + 1/312+ 13'12+ ,5i2)2 - 4,c~5 -/3'7i2 ) 2 b) o~,~ ~r ==~

-

sup{]c~ +

~1, I ~ - ~1}.

a) follows from Example 1.2.2.7 d). b) Take ~o,~b E IR such that o~ =

i~1~ ',o ,

~-

0 .= 2(r

-

i/~1~ 'r

and put ~).

Then I~ + 91 ~ =

rI~1~ ~

+ I~1~'r

~ = fl I~l • I~l~ ~

- I~l ~ + I~l ~ + 21~119J co~

so that 0 sup{Ic~ +/~12, Ic~ -/~12 } - I~1~ + I~l ~ + 21c~11911cos ~1. On the other side, ~_

9=1= _ i 1~12~=~o _ j~l~[~_

= I~14 + 1 9 1 4 -

121~ = +

21~1=1 = -

l l~j ~ _ 191=~,ol~ =

21~1~191~coso,

4 Io~2 - ~212=

o

1.2 Operators

= 4 (Iozl 4 + 2lo~l=l~l 2 + I/3l 4 -

85

la} 4 - I ~ l 4 + 2lc~l~-I~l 2 cos o) =

= 8[al2[/312(1 + cosO) -- 16Ic~[~[/3[2 cos 2

0

By a),

1(

- ~

-

21~1 ~ + 2 ~ + 4 1 ~ 1 1 ~ 1 1 c o s ~ 1

0 I~1 ~ + I~1 ~ + 21~11fll I cos ~1 - s u p ( l ~

0)

-

+ ill2,1~ _ ill2}.

c) follows from b). Remark.

Example

b) will be generalized in Example 7.1.4.10 h). 1.2.2.9

Take p C [1, cx~] U {0}. Given x e gP, define

xr'IN

a)

m

0 x(n-1)

>I[4,

n,

>

xe 9IN -----+ IK,

n,

> x ( n + 1).

/fn=l ifn~l,

x r , xg C gP and

for every x c gP. Define

b)

u~ 9gP

>

gP,

x~

>

xr

(the right shift on t?),

ui " gP

>

gP,

x,

>

x~

(the left s h i f t on gP).

u~, u~ e s

IlUrl] - 1, llU~]] = 1, U~ is injective, u~ is surjective.

c) u~ o ur is the identity operator on gP. d)

If a e ]K\{O} and x, y E gP such that

( a l - ur)x = y then x(n)-

1

~-~aky(k ) k=l

for all n E IN.

86

1. Banach Spaces

e)

al-

ur is injective for all a G IK.

All the above assertions hold for c in place of gP.

a), b), and c) are easy to see. d) We prove the assertion by complete induction. The assertion is obvious for n = 1. Let n C IN and assume the assertion holds for n. Then a x ( n + 1) - x(n) = y(n + 1),

so that x(n + 1) =

1 1 x(n) + - y ( n + 1) - ~ O~

n 1 n+l y ~ aky(k) + - g - ~ a y(n + 1) = k=l

1

n+l

= a.+2 Z

akY(k)"

k--1

e) follows from d).

m

E x a m p l e 1.2.2.10 ( 0 ) Let T be a locally compact space and let Co(T) be the set consisting of those elements of C(T) which vanish at the infinity (if T is compact then Co(T):= C(T)). Then Co(T) is a closed vector subspace of C(T) and Co(T)' can be identified with the Banach space .Mb(T) of bounded Radon measures on T . The proof can be found, for example, in N. Bourbaki, Integration, Ch. II (1952), w II Remark.

If T is endowed with the discrete topology then ~o(T) - C o ( T )

E x a m p l e 1.2.2.11 ( 1 ) (2) (3) If T is a completely regular space, then C(T)' can be identified with the Banach space of Radon measures on the Stone-Cech compactification of T . This is an immediate consequence of Example 1.2.2.10. E x a m p l e 1.2.2.12 set T .

m

( 0 ) gl(T), co(T)', and c(T)' are isometric for any

1.2 Operators

87

By Example 1.2.2.3 e) e I(T) and co(T)' are isometric. To show that el(T) and c(T)' are isometric, we may assume T to be infinite. Consider T whith the discrete topology and let T* be the Alexandroff compactification of T . Then c(T) can be identified with C(T*). Hence c(T)' can be identified with the Banach space Mb(T*) of bounded Radon measures on T* (Example 1.2.2.10). But Mb(T*) is isometric to gl(T*) and gl(T*) is isometric to t~l(T), since T and T* have the same cardinality. Therefore c(T) and t~l(T) are isometric. I

Remark.

co(T)' and c(T)' are always isometric even though co(T) and c(T)

are not isometric when T is infinite (Example 1.2.2.4 d) ). ( 0 ) Let (Et)tET be a family of normed spaces. Take

P r o p o s i t i o n 1.2.2.13

p and q weakly conjugate exponents. Define

tET

g-- {xtE

HE~ tET

I (IIx'tII)tETE ~q(T) },

and endow E and F with the norms

E

>XR+,

x,

>ll(Ix, ll):~rllp,

F

>n%,

x',

>ll(llx'~ll)~TIIq,

(Proposition 1.1.2.7 a)). Given x' E F , define x'' E

>]K,

x,

,

tET

a)

x' E E' and

I1~'11- IIx'll for

b)

If p r oc and q r O or if T is finite, then the map

~w~y x' E F .

>

F

E'

x'~

N ~ x'

is an isometry. a) Given x E E and x' E F ,

y ~ I<xt, x~>l < ~ tET

tET

IIx~ll Ilx~ll < Ilxll IIx'll

88

1. Banach Spaces

N

(Proposition 1.2.1.4 a), Example 1.2.2.3 a)), so that x' E E' and IIx'll < IIx'll.

Take x' E F and a E ]0, Ilx'l][. By Proposition 1.2.2.3 b),c), there is a z E gP(T) # such that z > 0 and

II~'~llz(t) >

~.

~-:~ I ~'~llz(t) >

~.

tET

Let S E ~3s(T ) with

tES

By Proposition 1.2.1.4 b), there is an (Xt)tET E l - l E t # tET

such that Xt - - O

for every t E T \ S

and

E(xt, x~t)z(t) :> a . tES

Put y := ( z ( t ) ~ ) t ~

E E.

Then 1

1

_< ( E z ( t ) P ) ~ tET

tET

if p E [1, oc[ and IlyI - supz(t)ilxt I <_ 1 tET

if p E {0, c~}. It follows (Proposition 1.2.1.4 b))

II~'ll _> ~'(y)- Z(y~, ~'~)= tET

_< 1

1.2 Operators

89

tET

Since a is arbitrary,

I1~'11 _> I1~'11,

IIx'll = IIx'll.

b) Take v~ C E ' . Let s E T . Given x C E s , let Y denote the element of E given by Y

t,

~

x

L 0

ift=s ift=fis

and define X s' " E~ !

> IK

x,

> O(Y).

!

Then x s C E s . P u t

x' :=

9 I-[ E;. tCT

Take y C IK (T) . Then

tET

tET

for every x C E with

IIx~ll < 1 for every t C T . Hence

tET

(Proposition 1.2.1.4 b)). Since y is arbitrary, x ' E F and

IIx' I < IlOll (Proposition 1.2.2.2), and x ' = 0 on

G := {x c E l{t z T Ix~ # 0} is finite}. Since G is dense in E (Proposition 1.1.2.7 c) ), we deduce that x' = 0 and

IIx'll < IlOll- IIx'll < IIx'll,

IIx'll- Ilx'll.

Hence the m a p F is an isometry,

)E',

x'.~

;x' i

90

1. Banach Spaces

C o r o l l a r y 1.2.2.14 Let T be an infinite set endowed with the discrete topology and let fiT denote the Stone-Cech compactification of T . Put E "- co(T) x C ( f l T \ T ) ,

F

:=

e'(T)

x Mb(flT\T)

and endow E and F with the norms

E

~ JR+,

(x,y),

F ---+ IR+,

(z,#),

sup{l]x[[~, []Y [}, ~1 zll, + ll,II

(Proposition 1.2.2.13 a)).

a) For (z,#) E F , define (z, ~) . E

~~ ,

(~. y) .

E

x(t)z(t)+

fvd,.

tET N

Then (z,#) e E' for every (z,#) e F and the map u . F

> E' ,

(z. . ) .

. (~. . )

is an isometry.

b)

For (z,#) e F , define

(z, #) 9C(flT)

) IK ,

X

E

x(t)z(t) + f x

[(flT\T)d#.

tET

Then (~,~) E Adb(flT) for every (z,#) e F (Example 1.2.2.10) and the map

is an isometry.

c) For y E g~176 denote by ~ its continuous extension to fiT and for # C M b ( Z T ) , put

Th~. ~ c ( e ~ ( T ) ) ' fo~ ~v~y ~ ~ M ~ ( g T ) and th~ map ~

is an isometry.

M~(gT)

> (e~

~,

~

1.2 Operators

d)

91

w o v o u - l : E' --+ ( t ~ ( T ) ) ' is an isometry. a) follows from Example 1.2.2.3 e), Example 1.2.2.10, and Proposition

1.2.2.13. b) is easy to see. c) follows from Example 1.2.2.11. d) follows from a), b), and c).

m

92

1. Banach Spaces

1.2.3 I n f i n i t e M a t r i c e s

D e f i n i t i o n 1.2.3.1 ( 0 ) An infinite matrix is a function S x T -+ IK, where S and T are sets. Let k : S • T -+ IK be an infinite matrix and take p C [1, oc] U {0}. Let q be the conjugate exponent of p. If k(s, .) c gP(T) for every s C S , then define

M

kx" S

~ IK,

s,

)E

k(s,t)x(t)

tET for x C gq(T) (Example 1.2.2.3 a)). If k(., t) C gP(S) for every t E T , then we define

u sES for x e gq(s) (Example 1.2.2.3

a) ).

If S and T are finite, then k is a matrix and kx is the usual multiplication of the matrix with a vector. D e f i n i t i o n 1.2.3.2

(0)

Let S , T be sets and take p,q e [1,oc] U {0}.

Let gP'q(s, T) denote the set offunctions k: S x T ~ ]K such that k(s, .) E ~.q(T) for every s c S and that

(Ilk(s, .)llq)scSE ~P(S) endowed with the norm gP'q(S,T)

>]R+,

k,

~ll(llk(s,.)[lq)sesllp

(Proposition 1.1.2. 7 a) ). We define ~o'q(s, T)

D

~

{k E ~~

It E T ~

k ( . , t ) C co(S)}

and endow fo'q(s, T) with the restriction of the norm of f~'q(s, T ) . By Proposition 1.1.2.7, gP'q(S,T) is a Banach space. ]K (sxT) is dense in fP'q(S,T) whenever p :/: oo and q :/: oo. It is easy to see that fo'q(S,T) is a closed subspace of t~'q(s, T), so that it is a Banach space, f~ T) is a closed subspace of to'q(s, T). Let k C IK SxT. Then k C t!2'2(S,T) iff

Z

(s,t)ESxT

Ik( ,t)l <

and in this case the above sum is I]kll2

1.2 Operators

Proposition 1.2.3.3

Let S, T be sets and take p,p', q, q' C p<_p',

[1, c~[

93

with

q<_q'.

Then ~P'q(s, T) c ~P"q'(S, T) A ~?'~

T) n e~

t~P'~ T) U t~~

t~~176 T),

T)

C

T) ,

and the various inclusion maps above have norm at most 1. m

The assertion follows from Proposition 1.1.2.6 b), e). Proposition 1.2.3.4

( 0 )

Let S , T

be sets. Take

p,r e [1, cc] U {0} and let q be the conjugate exponent of p.

a)

N

kx C

gr(s)

and A

Ilkxll, <_ Ilkll II~ll, whenever k ~ er'q(S, T) and x e ep(T).

b)

The map v) k'~P(T)

~ er(S),

x,

N > kx

belongs to f_.(gP(T), ~7"(S)) and N

Ilkll ~ Ikll w h e n e v e r k e ~r,q(s, T ) . I f r E {0, (3o} (resp. if r e {0, (3o} and p -- cxD)

then N

Ilkl = I kll

N

N

(reap ]ik I c0(T)ll- Ilkl]- Ilkl])

whenever k C gr,q( s, T) .

c)

For S ~ 0 and r - oc, the map A

is an isometry iff p or T is finite.

94

1. Banach Spaces

d) If p ~ oo, then the following are equivalent for every k 9 gr'q(s, T)-

dl)

N

k(e'(T)) c co(S)

d~.) k(., t) 9 co(S) e)

(~esp. c(S) ); (rasp.

c(S) ) for every t 9 T .

The map ~.o'I(S,T)

A

> E.(co(T),co(S)),

k, ~ ~ k

is an isometry.

f) (Woeplitz, 1911) Given k 9 go'l(S,T), the following are equivalent: A

fl) kz e c(S) fo~ ~w~y x e c(T), f~) E k(., t) e c(S). tET

g) If R is a set, u 9 s

and k 9 t~2'2(S,T), then there is a

unique h 9 ~2,2( R, T) such that N

N

h =uok; we have

ilhii < ii~li iiklla) Given s 9 S,

tET A

(Example 1.2.2.3 b), Proposition 1.2.1.4 a)). Thus kx 9 gr(s) and N

ilk~ilr < iikli il~ii~. b) By a), Proposition 1.1.6.11, and Proposition 1.2.1.1 d =v a, N

k 9 L(~,(T), e r(s)) and N

ilk i_< ilkll. Assume now that r 9 {0, oo}. By Example 1.2.2.3 b), c) and Proposition 1.2.1.4 b),

1.2 Operators

{n

_< ~up IIk~ll~

95

I 9 9 e(T)#) = Ilklln

for every s E S , and so n

Ikll_< Ilkll,

n

Ilkll = Ilkl.

Finally, assume t h a t r E {0, cx)} and p = cx~. T h e n

< sup

{nI]kx]]o~ Ix E co(T) # } =

]]k ]co(T)]]

for every s E S (Example 1.2.2.3 c), Proposition 1.2.1.4 b) ). Thus N

N

Ilkll <_ Ilk I co(T)ll <_ Ilkll- Ilkll (Proposition 1.2.1.4 c) ) and n

N

Ilk I ~o(T)tl- Ilkll = Ilkll. c) Assume t h a t either p or T is finite. Take u E s

k.S•

.~,

(~,t),

g~(S)). P u t

~(~T)(~).

Then

tET

tET

tET

= I(~x)(~)l _< I1~11~ < I1~11 I1~11. for every s E S , and x E IK (T) (Proposition 1.2.1.4 a) ). Thus

k(~, .)E eq(T) and

IIk(~,-)ll~ <_ II~ll for every s E S (Example 1.2.2.2). Hence k E t ~ ' q ( S , T ) . Since

96

1. Banach Spaces

n T

(k~,)(~) - k(~, t) -(u~T)(~) for every (s, t) E S x T , we have t h a t n T

ke t = ue T N

for every t E T . So k = u

I K (T) . Since ~ ( ( T ) is dense in t~P(T) (Proposition n 1.1.2.6 c) ) we deduce that k - u. Hence the m a p on

N

~.~176

~ ff_.(~.P(T), ~ ~ 1 7 6

k,

~k

is surjective. By b) it is an isometry. Assume now that p -

cc and that T is infinite. Take s E S and let ~ be

a free ultrafilter (i.e. an ultrafilter containing no singleton set) on T. Define u'e~(T) Then u E s

)/~(S),

x,

)(limx(~))e s.

u(eTT) = ess , and u -- 0

~(S)),

on

I N (T)

. Assume there

N

is a k E t~~ ' I ( S , T ) with k - u .

Then n T

k(~, t) = ( k ~ , ) ( ~ )

= (~,~T)(~) = o

for every t E T and we obtain the contradiction n

T

1 - (ueT)(s) = (keT)(S) = E

k(s, t)eT(t) -- O.

tcT

Hence the map

e~

---+ S ( e ~ ( T ) , t~176

N

k,

)k

is not surjective. dl =~ d2 holds since e T E t~P(T) and N

k(., t ) -

ke T C Co(S)

(resp. c(T) ).

M

d2 ::v dl . By d2), k(IK (T)) C co(S) (resp. c ( T ) ) . Since IK (T) is dense in t~P(T) (Proposition 1.1.2.6 c)), dl) follows from b) and Example 1.1.2.3. M

e) By b) and d2 ~ d l , given k c t ~ o ' l ( S , T ) , k

may be taken to be an

operator from co(T) to co(S). By c) and dl :::> d2), the map M

~ o " ( s , T ) --+ ~(~0(T), c0(S)),

k,

~k

1.2 Operators

97

is an isometry. fl ==~ f2. We have t h a t n

T

k(., t ) : k ~ c c(S). tET

f2 ::a fl. We may assume that T is infinite. Let ~ denote the filter of cofinite subsets of T , i.e. I e g3s(T)},

~ "= { T \ B

and put a " - lim x ( ~ ) . Then x -

ae T 6 c o ( T ) , so that by b) and d2 ~ d l , N

k(~- ~)c

~0(s).

Hence n

n

n

T

N

k~ = k(x - ~ 4 ) + ~ k ~ = k(~ - ~ 4 ) + ~ ~

k(., t) c ~(z).

tET

g) We set

h-R•

~,

N

(r,t), "(~k~T)(~).

Then N

2

E

2

t E T rE R

(r,t)EIRxT

tET

~

12--

tET

tET

sES

(Proposition 1.2.1.4 a)), so that h e tF'2(R, T) and

Ilhll __ I1~II Ilkll. We have N

N T

T

for all (r, t) C R x T , so that

n

2

98

1. Banach Spaces

n

T

n

T

he t = uke t

for all t E T. By Proposition 1.1.2.6 c), n

n

h=uok.

The uniqueness of h is obvious. Remark. Some of the results in Proposition 1.2.3.4 are actual special cases of the theory of integral operators, which will be treated in Section 3.1.6.

P r o p o s i t i o n 1.2.3.5

( "l )

( 2 ) Let R , S , T be sets and p,p' weakly conjugate exponents. Take n, q E [1, cr U {0}. Let n' be the conjugate exponent of n . Take k C gP'n'(S, T ) , h e ~.q'P'(R, S) , and let h o k. R • T --+ ~,

(~,

t),

~ ~ h(~, ~)k(~, t) sES

(Example 1.2.2.3 a)). Then h o k e gq,n'(R, T),

lib o kll _< llhll Ilkll, and n n n

h o kx = h ( k x ) whenever x e gn(T) . If k e ~ , 1 ( S , T) and h e Go" (R, S) , then

h o k e e~ ,~(n, s). Take x c t~n(T) and r C R. Then

(s,t)eSxT

sES

<_ ~

Ih(r,~)l IIk(~, .)ll~,lJ~Jl~ <_ lib(r, .)JJ~,lJklJJl~ll~

sES

(Example 1.2.2.3 a) ). It follows that

tET

teT

tET

sES

1.2 Operators

= E

h(r, s) E

sES

k(s, t)x(t)

)

._n N N h(r, s)(kx)(s) = h ( k x ) ( r ) ,

= E

tET

99

sES

Ih(~, ~)1 Ik(~, t)l Ix(t)l < tET

(s,t)ESxT

_< ]lh(~,-)]1~, ]]kll !1~11~. By Proposition 1.2.2.2, h o k(r,-)E en'(T),

[Ih o k(r, ")[In' <_ Ilh(r,')l[,'[lkll 9

Furthermore

(llh o k(~, ")11~')

rcR

9 ~(R),

h o k e ~'~'(R, T),

Ilh o kll < Ilhll Ilkl].

By the above result, N N

N

h o k(x) = h(kx).

The last assertion now follows from Propositiop 1.2.3.4 d2 ==~dl.

I

Remark. If R,S and T are finite, then h o k is the usual product of the matrices h and k.

Proposition 1.2.3.6

( ~ ) ( 2 ) ( ] ) Let S , T be sets. Let p,p' and

q, q~ be weakly conjugate exponents.

a) (k(s, t)k'(s,t))(s,t)ES•

is summable whenever k E gP'q(S,T) and k' E g~"q'(S,T) .

Given k' E gP"q'(S, T ) , define k " ~P'q(S, T)

~ IK ,

k.

,~ ~

k(~,t)k'(~,t).

(s,t)ESxT

b) k'E ~P,q(S,T)' and [Ik'[I = ]lk'l]. c) IS p ~ c~ and p' ~ O, then the map ~,',~' (s, T )

is an isometry.

~ ~,,~(s, T ) ' ,

k' ,

~. k"

100

1. Banach Spaces

Given s E S , k(s,.) E e~(T) and k'(s,.) E # ( T ) a e~(T)' (Example 1.2.2.3 b) ). The assertions follow from Proposition 1.2.2.13 (and Example 1.2.2.3 d), e)). I

,,o,ositio,

( 1 ) (

2

)

Stone-Cech

compactification of T with respect to the discrete topology on T , and .A4 the Banach space of Radon measures on Z T . Take p E [1, cx~[ U{0}. Let q be the conjugate exponent of p and E the set of maps #" S ~ Ad with

(11~11)~ E e~(s). Extend each y E g ~ ( T ) to a continuous function on ~ T , which we denote with the same symbol. Given # E E , define

~.e',~(S,T)

~ X,

k,

~~/k(~,.)d,~ sES

(Example 1.2.2.3 a)).

a) E is a vector subspace of M s and the map

is a compiete norm. Endow E with this norm.

b)

~ E e~,~(s, T)' ~h~n~wr . 9 E and th~ ,nap

E

~tP'~(S,T)',

~,

~

is an isometry.

The assertions follow from Proposition 1.1.2.7 and Proposition 1.2.2.13 (and Example 1.2.2.11). I

Proposition 1.2.3.8

( 1

) (3)

n~t S,T ~ ~t~, p,q ~aktu conj~gat~

exponents, and take r E [1, c<)]. u

a) kx E gr (T) and u

Ilkxll~ _< Ilkll Ilxll~ whenever k E gq,r( S, T) and x E gP( S) .

1.2 Operators

b)

101

Given k E ~q,r(S,T), the map t.)

k-e~(S) belongs to s

>er(T),

U

x,

~ kx

e ~(T)) and U

IIkll < IIkll. c) /f p - - 1 and

q = oc, then the map

e~,'(S,T)

, L <(s),e~(T)),

U

k,

~k

is an isometry.

d)

/f r = oc and p r ec then the following are equivalent whenever k e e~,~(s,T)

9

u

u

d~) k(s

C co(T)

d2) k(s,.) E co(T)

(resp. k(eP(S)) c c(T)); (resp. k(s,.) E c(T)) for every s E S .

a) Let r' be the conjugate exponent of r. Then

tET

sES

tET

tET

sES

sES

~-- E Ix(s)lik(s' sES

tET

")llrlly Ir' --~ IlYllr'llkll Ilxl p

u E t~r (T) whenever y E ] K (T) (Proposition 1.1.6.11, Example 1.2.2.3 a) ). Thus kx and U

Ilkzll~ <_ Ilkll llxll, (Proposition 1.2.2.2). b) follows from a) and Proposition 1.2.1.1 d ~ a (and Proposition 1.1.6.11). c) The map ~~176

~ ~(~I(s),

~r(T)),

k,

u } k

102

1. Banach Spaces

is linear (b) and Proposition 1.1.6.11). Take k 9 f~,r(s, T). Then, for every s 9 S, u

U

u

S

Ilkll = Ilkll II~ II, > IIk(~, ")11~ (Proposition 1.2.1.4 a) ) and so U

Ilkll_

sup sES

IIk(~,.)ll~ = Ilkll.

U

By b), Ilkll = IlkllTake u 9 ~(e'(S),e~(T)). Put k" S x T

> IN:,

(s,t),

,~ (ueS)(t).

Then

k(s, . ) = uef e e~(T) and

IIk(~, ")llr

=

II~fll~ _< I1~11

whenever s 9 S . Thus k 9 t~~'~(S, T ) . Given s 9 S , u S kG - k ( s , ' ) - ue s , U

a n d s o ku = u

on IK (s) .By b), k = u ,since IK (s) is dense in ~1 (S) (Proposition

1.1.2.6 c)). dl =~ d2 follows from esS 9 fp(S) and the fact that U S

k(~, . ) - k~ e co(T) (e c(T)) for every s E S.

u

d2 =~ dl. Given x e ]K (s), k x 9 co(T)

(e c(T)) and dl) follows from b)

and Proposition 1.1.2.6 c) (and Example 1.1.2.3). Remark.

I

If p ~= 1, then the map u

~q"(S,T)

>s

k.

>k

need not be surjective. Indeed, if S - T and 1 < p < r , then the embedding u

gP(S) c K ( S ) is not of the form k.

1.2 Operators

Definition

1.2.3.9

1 )

(

103

A family (A~)~e, of sets is called disjoint if A~ N A x = 0

for any distinct t, A E I . Proposition

1.2.3.10

(

1 ) Let S,T

be sets and k a function S x T --+ IK

such that k(s, .) E/~I(T) for every s 6 S . Then the following are equivalent: M

a)

kx e g~176 whenever x e g ~ ( T ) .

b)

kx e e~(S) So~ ~r

c)

ke s 6

N

d)

n

T

~cx~

9 e Co(T).

(S) for every countable subset B of T .

There is a sequence (C~)n~IN in IR+\{0} with the property that for every disjoint sequence ( B n ) n ~

in q3s(T ) there is a sequence (an)ne~ in IR N

such that a,, > c,~ for every n 6 IN and that kx C f ~ 1 7 6 the map given by 0

x'T

> IK,

t,

if

nEIN

if t 6 Bn.

an e)

t6T\ U

)

k 6 g~'~(S,T).

If these conditions are fulfilled, then the map Cl

N

k e ~ ( T ) --+ e~176 is in s

~(S))

x:

~,kx

and M

M

[Ikl] = Ilk I c o ( T ) [ I - sup IIk(s,.)[I,. sES

a :=> b and a :=> c are trivial. 1 b ==~ d. Given n E IN, put cn .._-_ ~.

c ::=> d. Given n r IN, put cn - 1. d => e. Given t c T , N T

k(, t) = kr Define

e ~(s)

Bn

where x is

104

1. Banach Spaces

kl:SxT

>IR,

(s,t),

;sup{rek(s,t),0}

k2:SxT

>IR,

(s,t),

>sup{-rek(s,t),O}

k3:SxT

>IR,

(s,t),

>sup{imk(s,t),0}

k4: S x T

> IR,

(s,t),

>sup{-imk(s,t),0}

Assume that (Ilk(s, ")lll)s~S is not bounded. Then there is a j 9 {1, 2, 3, 4} such that (llkj(s,.)ll~)ses is not bounded. P u t B0 -

0 and construct a sequence

(sn)ne~ in S and an increasing sequence (Bn)ne~ in 913I(T) inductively such that n

Sn r {Sk I k 9 I N n - l } ,

E kj(sn, t) > - En tEBn\Bn-1

for every n 9 IN. Take n 9 IN and assume that the sequences have been constructed up to n -

1. Then there is an Sn 9 S~k{Sk I k 9 I N n - l }

such that

Ilkj(s~,.)llx >

n

Cn

+ 1 + (CardBn_l)

sup IIk(.,t)ll~.

tE B,~-1

There is a finite subset B~ of T with B~-I C Bn and

E

kj(sn, t)< 1.

tET\Bn Then

E kj(sn, t) tCBn\Bn-1

Ilkj(Sn, .)Ill -

kj(sn' t) -

E

teBn- 1

n

Z kj(sn' t) > Cn tCT\Bn

This completes the inductive construction. We put

cn . - {t e B.\B~_~ I kj(~,~, t) > 0}. Then (Cn)nc~ is a disjoint sequence in q3l(T ) . By d), there is a sequence N

(an)nelN in IR such that an > ~,~ for every n 9 IN and kz 9 g~176 where

x'T

>IK,

t,

0

if t c T \

)

U cn nEIN

an

if t c C n ;

1.2 Operators

105

Given n E IN,

n

i

i

I(k~)(~.)l = ~

k ( ~ , t)x(t) >_ E

tET

E

~ O~n

kj(sn, t)x(t) >_

tET

tE Bn \ B n - 1

k j ( s n , t ) > oLn - n ~ n , gn

n

which contradicts kx C g~ ( S) . e ===>a, and the last assertion follow from Proposition 1.2.3.4 b). Theorem

1.2.3.11

(

1)

I

(Kojima-Schur) Let S, T be sets and k" S • T -+ IK

a ]unction with k(s, .) C gl(T) for every s E S . Then the following are equivalent: M

~) k~ e c(S) /o~ ~ y b)

x e c(T).

k(-,t) E c(S) for every t c T , E k(-,t) C c ( S ) , and k C g~c'I(S,T). tET

If these conditions are fulfilled, then the map

~(T) is in s

>~(S),

M

~,

>kx

Ilkll.

c(S)) and has norm

a =~ b. Given t E T , e T and e T belong to c(T) and N T

k(., t ) - k~ ~ c(S),

N T

Z k(., t)= k~ c ~(s). lET

By Proposition 1.2.3.10 b =~ e, k E f ~ ' I ( S , T ) . M

b =~ a. By Proposition 1.2.3.4 a),b), kx E g~(S) whenever x c g~(T) and the map

e~(T)

~e~(s),

x,

is continuous. It is obvious that

By continuity, n

k(co(T)) c c(S).

M

~ kx

106

1. Banach Spaces

There is an a 9 IK with x -

s e T 9 co(T). Since

k4 - Z

k(., t) c c ( S ) ,

tET M

kz c c ( S ) . The final assertion follows from the last assertion of Proposition 1.2.3.10.

I Remark.

The above theorem still holds for co(S) in place of c ( S ) .

T h e o r e m 1.2.3.12

1)

(

(I. Schur) Let S , T be sets and k" S • T -+ IK a

function with k(s, .) E gl(T) for every s E S . Then the following are equivalent: N

a)

kx e c(S) for every x 9 t ~ 1 7 6

b)

ke s 9 c(S) for every countable subset B of T .

M T

c) k(.,t) c c(S) .for every t O T ,

and for any e > O there is a finite subset

B of T such that

teT\B

for every s E S . Assume that k(., t) e c(S) for every t e T . Define e ~S(S)},

a "- { S \ A I A

a'T

~IK,

t,

~limk(-,t).

If the above conditions are fulfilled, then a C gl ( T ) ,

lim Ilk(s,- ) -all~ = 0 , s,~ the map N

M

k . e~ ( T ) ~

N

c(s),

~,

) k~

A

kll = Ilk I co(T)ll = sup IIk(~, )Ill sCS

1.2 Operators

107

and N

a(t)x(t),

limkx = ~

tcT for every x C g ~ ( T ) . a =~ b is trivial. n b =~ a. Let B be a subset of T . Assume that ke T ~ c(S). Then there is a sequence (s~)ne~ in S with

n T n T lim inf I (keB)(Sn) --(keB)(Sn+l ) [ > O. Given n c IN, put

Ca "= { k ( ~ , ") # 0}

n

B

and

c:=Uc,. ncIN

Then C is countable and n T

n T

(keB)(sn)-(kec)(Sn)

for every n E IN. Thus

I

NT lim inf I (kec)(Sn) - ( k eNcT) ( S n + l ) > 0 n T

n T

which contradicts the fact t h a t ke c C c(S). Hence ke B c c(S). n By Proposition 1.2.3.10, kx c g~(S) whenever x c g~(T) and the map

e~(T)

~e~(s),

~~

n

k~

is continuous. Since the vector subspace of e~(T) generated by {e T I B C T} n is dense in g~(T) (Proposition 1.1.2.6 d)), kx C c(S) for every x c g~(T) (Example 1.1.2.3). a =~ c. Given t c T , eT E ~ ( T )

and n T

k(., t) - k ~ e ~(S). Take c > 0. Assume t h a t for every finite subset B of T there is an s E S with

108

1. Banach @aces

Ik(,, t)l > c.

tET\B P u t Bo "= 0 and construct a sequence (Sn)neIN in S and an increasing sequence (Bn)nelN in ~I3s(T) inductively such t h a t s

8n ~ {Sk ] k C I N n - l } ,

E Ik(sn't) - a(t)l < -6' tCBn-1 5s

~ Ik(sn, teT\B,~

Ik(,~,t)l > 6 '

teB,~\Bn-1

t)l

<

s

for every n C IN. Take n C IN and a s s u m e t h a t the sequences have been cons t r u c t e d up to n -

1. Since lim k(., t) -- a(t) a

for every t E B n - 1 , there is a finite subset A of S with

Ik(~, t)

s -

a(t)l

<

tCBn-1 whenever s C S \ A .

Put

C := A u {sk I k e IN,_~}. Given s c C , there is a finite subset Ds of T with

tET\D~

Ik(,,t)

< ~.

Define

(U sCC

T h e n D is a finite subset of T and

t6T\D for every s C C . By the above a s s u m p t i o n , there is an sn C S \ C

E

Ik(s~'t)[ -> c.

tCT\D Let Bn be a finite subset of T with D C Bn and

with

1.2 Operators

109

C

Ik(~, t)l < g.

tET\Bn Then

Ik(sn, t) l = tc B,~ \ B n - ~

~

> ~

tk(Sn, t)l--

tET\D

~

Ik(~,t)l-

tCT\ Bn_ ~

~

Ik(sn, t)l _>

tCT\ Bn

c

Ik(s~,t)J >e

5c

= 6 6

teT\B~

T h i s c o m p l e t e s the i n d u c t i v e c o n s t r u c t i o n . W e now define a m a p x " T - - + IK. W e set x - 0 on

(~\ U ~o)u (U (~.-,\~-~/) nEIN

G i v e n n E IN and

nEIN

t c B2n\B2n-1,

we set

x(t) := k ( ~ , t)l if

k(s2,, t) -r O,

x(t) - 0

and

otherwise. Then x e e~(T)

I(~x)(s2~ (;x)(~2n-,)l-IEk(s2o,~)x(,)- E tET

>_

~

tCB2n \ B 2 n - 1

tEB2n-2

tE B2n \ B2n- 1

-I E

k(~2n-"~)x(')I->

tET

k(~2n,~lxl'll-I ~ (k(s2n,'/- k(s2n ,,,~)x(,~ I-

tCT\ B2n

tEB2n\B2n-1

and

I~(s2~,,~l - ~

~

tCB2n-2

-~ tET\B2n

k(s2~

tcT\B2n

Ik/s2n,,t-o~,/I - ~

I~(s2n,~/I-

tEB2n-2

~ tcT\B2n-1

Ik~2n 1,'~-a/'/I -

I~(s~o-l,'~l>

110

I. Banach @aces

5g"

E

> ~--2g-2g=g

E

E

M

for every n 9 IN, which contradicts the fact that kx 9 c(S). c ==> a & the last assertion. Take e > 0. By c) there is a finite subset B of T with

Ik(~, t)l < E tET\B

whenever s 9 S . We obtain successively that

Z

la(t)l _< lim

inf

tET\B

Z Ik(~,t)l <

a 9 el(T),

tETkB

Ik(s, t) - a(t)[ <_

IIk(~,-) - allt = ~ tET

<-EIk(s't)-a(t)l+

Ik(s't)l + E

E

tCB

tET\B

la(t)l < E I k ( s , t ) - a ( t ) l + 2 e

tET\B

tEB

for every s 9 S , lim sup Ilk(s, .) - all1 < 2e, s,~'

and lim sup [[k(s,-) - ai], = 0 s,a

since e is arbitrary. Take x C e~176

Then

tET

tET

whenever s E S . Thus lim (kx)(s) - E s34

a(t)x(t)

tET

= 0

'

lim(~cx)(s) = E s,~i

$ET

a(t)x(t)

'

M

kz c c(S).

By the last assertion of Proposition 1.2.3.10, the map M

M

k:e~(T) is in s176

~c(S),

z,

>kz

c(S)) and N

N

Ilkll = Ilk l c o ( T ) l l - sup Ilk(s,-)I,. sCS

I

1.2 Operators

C o r o l l a r y 1.2.3.13

1)

(

L~t r

111

b~ an arbitrary and S an infinite set. Let

:= { S \ A I A e q3S(S)}, and let (xs)ses be a family in gl(T) such that

lime s,~

xs(t)

tEB

ezists whenever B is a countable subset of T . Define

x'T

;IK,

t,

>limxs(t).

xlll

=

s,~

Then x E ~1 (T) and

limllxs

-

0.

s,N

We define k" S x T

>IK,

(s,t),

>xs(t).

n T

By hypothesis, ke B E c(S) for every countable subset B of T and the conclusion follows from Theorem 1.2.3.12. C o r o l l a r y 1.2.3.14

(

1)

I

Let S , T

be sets and k a function S x T --+ IK

for which k(s, .) C gl(T) for every s C S . Then the following are equivalent:

a)

k e e~

b)

kx c co(S) fo~ ~ y

c)

k(.,t) e Co(S) for every t e T , and given c > O, there is a finite subset

M

x c e~(T)

B of T such that

tET\B

whenever s C S .

a =v b follows from Proposition 1.2.3.4 a). b ::v c. Given t E T , etT C g~

so that M T

k(., t ) = k~ e c0(S) By Theorem 1.2.3.12 a =a c, given c > 0, there is a finite subset B of T such that

112

1. Banach Spaces

Ik(s,t)l

< c

tET\B

whenever s E S. c =~ a. Let c > 0. By c) there is a finite subset B of T with

Z

Ik(~, t)l <

c

tET\B

whenever s E S . Let n be the number of elements of B . Given t E B , there is a finite subset At of S with

Ik(~,t)l <

2n+1

whenever s E T\At. Put

A-

UAt . tEB

Then A is a finite subset of S and

II~( S, ")]I1 -- ~ tET

[~(S, t)] = ~

Ik(s, t)] + E

tEB

for every s E T \ A . Hence (Ilk(s, ")1[,)

tET\B

sES

I k ( s ' t)] < n

E

E

2n+1

E Co(S) and k E e~

T).

1.2 Operators

113

1.2.4 Q u o t i e n t Spaces

Definition

1.2.4.1

( 0 ) Let E be a vector space. Take A , B C V ( E ) , z

C E,

and a E IK. Define

A+B:={x+yl(x,y)

eAxB},

aA := {c~x l x C A } , z+A:=A+z:=A+{z}. Let F be a vector subspace of the vector space E . Given x, y c E , define x,,oy: ~ Let E / F

x-ycF.

denote the set of equivalence classes of ~

and a C IK, X + Y

and a X

belong to E / F .

Then given X , Y C E / F

E/F

is a vector space with

respect to these operations, with F as the null element. The dimension of E / F is called the codimension of F in E . The map E --+ F which maps each point of E into its equivalence class is linear and it is called the quotient map.

Unlike the dimension of F , which is intrinsic to F , the codimension depends on the vector space in which F is embedded, as well as on F . T h e o r e m 1.2.4.2 ( 0 ) Let F be a closed vector subspace of the normed space E and q: E --+ E / F the quotient map. Given X c E / F , define

I]Xll := inf [Ix]]. xEX

a)

The map E/F

>JR+,

x,

>llXll

is a norm, called the quotient n o r m of E / F .

E/F

endowed with this

norm is called the quotient space of E with respect to F .

b)

q is continuous and open (i.e. maps open sets into open sets) and IIq][ = 1 if E C F .

c) A subset g of E / F d)

A map f continuous.

of E / F

-1

is open iff q(U) is open. in a topological space T is continuous iff f o q

is

114

1. Banach Spaces

e) E / F is complete whenever E is complete. If F and E / F are complete, then E is complete.

f)

a) Take X , Y E E / F and a E IK. Take x E X and y E Y . Then

x + y E X + Y , ax E a X , so t h a t

IIX + Yll ~ IIx + yll ~ Ilxll + Ilyll,

II~Xll ~ IIo~xll = Io~1Ilxlt.

Since x and y are arbitrary,

IIX + YII ~ IlXll + IIYII,

II~Xll ~ I~1 IlXll.

For a r IlXll =

l(~x)ll

I1~

_< ~

Now suppose IlXll

1

= O.

IIo~Xll,

I~lllXll <_ II~Xll

,

IIo~Xll = Io~llXll

Then there is a sequence ( ~ ) . ~

to O. Since F is closed, so is X . Hence 0 E X , i.e. X -

.

in X converging F and X -

0 in

E/F. b) Given x E E , IIq~ll _< II~ll.

Hence q is continuous (Proposition 1.2.1.1 d =~ a) and

IIqll _< 1. Assume

E -r F . Then we can find X E E / F , X r F, and we have t h a t IlXll = Ilqxll-< IIqll IIxll for all x E X . Thus

]]xl] ~ Ilq]] IIXll,

1 _~ ]]ql],

Now take an open set, U, of E

]lq]] : 1.

and take x E U . For some r > 0,

UE(x) C U. Take Y E E / F with IiYI] < e. There is a y E Y with IlyII < e. Thus

x + y E US(x) c U,

qx + Y = q(x + y) E q(U).

Hence

u~/~(q~) c and q(U) is open.

q(U),

1.2 Operators

i 15

-1

c) If U is open, then

q (U) is open as well since q is continuous by b).

-1

Now assume that

q (U) is open. Then

is open, since, by b), q is an open mapping. d) If f is continuous, then f o q is continuous as well. Now assume t h a t f o q is continuous and let U be an open set of T . Then -1

q

U

= f o q(U)

--1

is open. By c),

f (U) is open. Hence f is continuous.

e) Let (X~)~eI be an absolutely summable family in E / F .

For ~ E I take

x, C XL with

IIx~ I_< 211X~ll 9 Then (x~)~ei is an absolutely summable family in E . By Corollary 1.1.6.10 a :=~ b,

(x~)~ei is summable. By Proposition 1.2.1.16 (XL)LCI is summable.

Thus by Corollaryl.l.6.10 b ~ a, E / F

is complete.

f) Let (x~)~er~ be a Cauchy sequence in E . By b), (qxn),~eIN is a Cauchy sequence in E / F .

Since E / F

is complete there is an x E E such that (qXn)ne~

converges to q x . Hence (q(Xn - X))nC~ converges to 0. For every n C IN let y~ C F such that

y~ll

<

IIq(xn

-- x -- y~)

--

(Xm

[IXn -- X-

-

x) l + -

1 n

Then IlYm --

Y n l l - II(xn

--

9 -- Ym)

]Xn - x - Y~]I + IIXm -- X -- Yml] +

+

(Xm -- ~n)ll --<

Xm -- xnll

1 1 <_ ]q(Xn -- x)II + - + Iiq(xm - x)[I + - + Iixm - xnll TL

for all m , n

m

C IN, so that (Yn)nc~ is a Cauchy sequence in F . Since F is

complete (Yn)ne~ converges to a y C F . It follows that (Xn)ne~ converges to x + y. Hence E is complete. Remark.

I

If G is a closed vector subspace of E it does not follow that q(G)

is closed even if E is complete (Example 1.2.4.15 d) ).

116

1. Banach Spaces

C o r o l l a r y 1.2.4.3

( 0 )

Let F be a closed vector subspace of the normed

space E and G a finite-dimensional vector subspace of E . Then F + G is a closed vector subspace of E . Let q : E ---+ E / F

be the quotient map. Then q(G) is a finite-dimensional

vector subspace and hence a closed vector subspace of E / F

(Corollary 1.2.3.6).

The conclusion now follows from -1

F + G = q (q(G)) and Theorem 1.2.4.2 b).

Remark.

m

In the above corollary we may not replace the above hypothesis "G

is finite-dimensional" by the weaker one "G is closed". A counterexample may be found in Example 1.2.4.15 e). C o r o l l a r y 1.2.4.4

Let F be a closed subspace of the normed space E . E is

separable iff F and E / F

are both separable.

It is clear t h a t if E is separable, then both F and E / F So assume t h a t F and E / F sets of F and E / F

are also separable.

are separable. Let A and B be countable dense

respectively. Let q : E ~ E / F

be the quotient map and

let f be a m a p of B in E with

q(f(X))=X for every X C B . Take x C E and c > 0. There is a Y C B such that Ilqw- Yll < c. Then Ilq(x - f (Y))]I = ]lqx - q(f(Y))l] = [Iqx - YI] < c. -1

x-f(Y)-A

is a dense set of q ( q ( z - f ( Y ) ) ) . H e n c e t h e r e i s a

yEA

with

] ] x - f (Y) - YI[ < e. Since x and G are arbitrary, A + f ( B ) is a dense set of E . But A + f ( B ) is countable, and so E is separable. Definition 1.2.4.5

linear map. Define

( 0 )

Let E , F

9

be vectorspaces and u" E - + F be a

1.2 Operators

Ker ~ -

~1(0)

(th~ k ~ . a

117

oS ~ ) ,

Im u := u(E)

(the image of u ) ,

Coker u := F / I m u

( t h e c o k e r n e l o$ u ),

:u is a vector subspace of E and Im u is a vector subspace of F .

D E {0} U [1, cxD] and

u r , ut are the right and the left shift of t~p,

ively, then Kerur = {0},

Imu~ = {x r ~P Ix1 = 0},

Keru~ = {x E ~ P l n E IN\{1} : = ~ xn = 0},

Imu~ = ~P.

a 1.2.4.6

( 0 )

E -4 E / G

the quotient map, and u : E - 4 F

Let E , F

be vector spaces, G a vector subspace of

T h e n there is a unique map v : E / G he f a c t o r i z a t i o n o f u t h r o u g h E / G .

-4 F

a linear m a p which is 0

such that u = v o q .

v is

v is linear and it is injective iff

: G . I n this case the map E/G

>Imu,

x,

>v x

tire and is called the a l g e b r a i c i s o m o r p h i s m

a s s o c i a t e d to u .

uniqueness of v follows from the surjectivity of q. Take X E E / G yEX.Then

x-yEG, ux-

andso

uy = u ( x -

remark, the m a p v : E / G

ux = uy.

-4 F , defined by p u t t i n g vX

--- E / G ,

y) = O,

:-- ux

where x is any element of X , is well-defined. Clearly, voq=u.

:, Y E E / G

and a , / 3 E IK. Take x E X and y E Y . T h e n a x + 8Y r a X + f l Y ,

v(o~X + f l Y ) = u(o~x + fly) = a u x + fluy = a v X + f l v Y ,

v is linear. ume t h a t K e r u = G . Take X E E / G

with v X = 0. T h e n

118

1. Banach Spaces

ux=vX

=0

for every x E X . Hence X C Ker u = G , so t h a t X = G . T h u s v is injective. Now suppose t h a t v is injective. T h e n given x C Ker u , v(q~) = v o q(~) - ~

= o,

so t h a t

qx=O,

i.e. x E G .

Hence Ker u = G . Proposition

I

1.2.4.7

( 0 )

Let E , F

be normed spaces, G a closed vector

subspace of E , q" E --+ E / G the quotient map, and put

~ - : : {u 9 Z:(E,F) Iu I G : 0}. .T is a closed vector subspace of s s

F) and the map

F) --+ .T ,

v,

>v o q

is an isometry. It is obvious t h a t 9v is a vector subspace of s

Take u c 9r

and

x C G . There is a sequence (un)ne~ in ~ converging to u. Given n E IN, Ilux-

~xll = I1(~- ~)xll ~ Ilu- u~ll Ilxll

(Proposition 1.2.1.4 a) ). Hence

ux = lim unx - O . ~ ---~ o o

Thus u restricts to 0 on G and so u c 9v . Hence ~" is closed. The m a p C(E/a,

P)

~ 7,

v,

) ~ o q

is obviously linear and

IIv o qll ~ Ilvll I[ql] <-IIvl] whenever v E s

(Corollary 1.2.1.5, T h e o r e m 1.2.4.2 b ) ) . Take u E

3c and let v be the factorization of u t h r o u g h E / G

X E E / G and x E X . T h e n

( L e m m a 1.2.4.6). Take

1.2 Operators

119

IIvXII = IIv(qx)ll = Iluxll ~ Ilull Ilxll, so t h a t v 9s

IlvXll ~< Ilull IlXll,

and

Ilvll ~< Ilull,

Ilvll = IIv o qll

(Proposition 1.2.1.1 d ==> a ) . T h e above m a p is therefore an isometry. 1

Proposition

1.2.4.8

~

0 ) Let E , F

I

be normed spaces and u " E - + F a

linear map. I f E is finite-dimensional, then u is continuous. First assume t h a t u is injective. We put

p:E

>~+,

x,

>ll~xll-

Then

p(x + y) = It~(x + y)ll = II~x + uyll ~ II~xll + Iluyll = p(x) + p(y), p(o~x) --II~(o~x)ll- I I o ~ x l l - I~1 I l u x l l - Io~lp(x), (p(~) = o) ~

(ll~xll --- o) ~

(~x = o ) ~

(x-

o)

whenever x, y C E and c~ E IK. Thus, p is a norm. By Minkowski's T h e o r e m ( T h e o r e m 1.1.3.4), there is a fl > 0 such t h a t

p(x) ~ ~llxll for every x E E . Thus

Iluxll = p(x) ~ ~llxll for every x C E , i.e. u is continuous (Proposition 1.2.1.1 d :=~ a). Now let u be arbitrary. Ker u is a finite-dimensional vector subspace of E and hence closed (Corollary 1.1.3.6). Let q:E

> E/Ker u

be the quotient m a p and v the factorization of u t h r o u g h E / K e r u .

Then

v is injective ( L e m m a 1.2.4.6). Since E / K e r u is finite-dimensional, the above considerations show that v is continuous. Thus u = v o q is also continuous.

I

120

1. Banach Spaces

C o r o l l a r y 1.2.4.9

( 0 ) If E, F are finite-dimensional normed spaces, then

every bijective linear map E ~ F is an isomorphism, C o r o l l a r y 1.2.4.10

m

The dual and the algebraic dual of the finite-

( 0 )

dimensional normed space E coincide. In particular, the dimension of E and E' coincide,

m

C o r o l l a r y 1.2.4.11

Let E , F

( 0 )

be normed spaces and u" E --~ F be

a linear map. If K e r u is closed (e.g. finite-dimensional) and I m u is finitedimensional, then u is continuous. Let q : E --+ E / K e r u be the quotient map and v the factorization of u through E / K e r u . Then v is injective (Lemma 1.2.4.6). Since

v ( E / K e r u) = Im u, E / K e r u is finite-dimensional. Hence v is continuous (Proposition 1.2.4.8) and so u = v o q is also continuous. If Ker u is finite-dimensional, then it is closed by Corollary 1.1.3.6.

Corollary 1.2.4.12 ( 0 ) The following are equivalent: a)

x' is continuous.

b)

Ker x' is closed.

c) x ' = 0

a

m

E

or Kerx' is not dense.

a => b =:> c is trivial. b => a follows from Corollary 1.2.4.11. c => b. Assume that Kerx' is not closed. Then Kerx' has a point of adherence x which does not belong to Ker x'. Then given y E E , 9'

( xy '-t x )-X='-(- Y~ ) )

-

x' ( y ) -

z , - ~ x C Ker

~'(y) ~

x-~X'(Y)x' ( ~ ) - o ,

C Ker x',

z'(y)

(Corollary 1.1.5.4), E C Ker x', which is a contradiction. Hence Ker x' is closed.

1.2 Operators

121

Every infinite-dimensional normed space contains a den-

C o r o l l a r y 1.2.4.13

se proper vector subspace. Let E be an infinite-dimensional normed space. By Corollary 1.2.1.2, E admits a discontinuous linear form x'. By Corollary 1.2.4.12 c ==~ a,

Ker x' is

a dense proper vector subspace of E . Example

II

Let /3T be the Stone-Cech compactification of the dis-

1.2.4.14

crete space T . We put A := ~ T \ T

and consider every x C g~

extended

continuously on ~ T . Put u : e~(T)

~ C(a),

~ ,

~ xla.

Then the factorization e~176

~ 6(A)

of u through e~(T)/co(t) is an isometry. By L e m m a 1.2.4.6 and Tietze's theorem, the factorization

e~(T)/co(T)

~ C(A)

of u is bijective and it is easy to see t h a t it is also n o r m - preserving. Example

1.2.4.15

I

Take E:--~ 1 XCo,

F := {(~,~) I 9 c e~}, G:={(x,0) u :E

~ co,

lxefl}, (z,y)

,

)y--x,

and endow E with the 1-norm.

b)

F = Keru.

c)

The factorization of u through E / F

d)

is an isometry.

G is a closed vector subspace of E but q(G) is not closed in E / F . q : E -+ E / F

is the quotient map.)

(Here

122

1. Banach Spaces

-1

e)

F + G = q (q(G)) = { ( x , y ) l x, y 9 gl} is not closed.

a) u is linear and

I1~(~, y ) l l -

I l y - xllo ~ Ilyllo + Ilxllo ~ Ilxll~ + Ilyllo - I I ( x , y)ll~.

b) is obvious. c) By L e m m a 1.2.4.6, the factorization of u is bijective. Take z C co and (x, y) E E with

z = ~(~, y) = y - ~.

By a), Ilzllo ~ II(y,x)ll~. It follows from z = u(O, z) and tlzllo = I1(o, z)ll~ that the factorization is an isometry. d) By c), we may identify E / F

with co. W i t h this identification,

q(C) = e 1 .

e) is easy to see.

I

1.2 Operators

1.2.5 C o m p l e m e n t e d

Subspaces

Proposition 1.2.5.1 E.

123

( 0 ) Let F and G be subspaces of the normed space

Then the map F x G

~, E ,

( x , y ) ~-+ z + y

is linear and continuous.

It is obvious t h a t the above m a p is linear. If we endow F • G with the 1-norm of the product then

ll~ + yll _< Ilxll + Ilyll = ll(x, y)ll~ proving continuity (Proposition 1.2.1.1 d =~ a).

Proposition 1.2.5.2 E,

q: E -+ E / F

( 0 )

Let E

I

be a normed space, F , G

subspaces of

the quotient map, and put u:F•

(x,y)

>E,

v :G

~, E / F ,

y~

>x+y,

> qy.

Then the following are equivalent:

a)

u is an isomorphism.

b)

F and G are closed in E and v is an isomorphism.

c)

F is closed in E and v is an isomorphism.

a ::v b. F x {0} and {0) x a

are closed sets of F x G . Since u is a

homeornorphism, F and G are closed sets of E .

q is continuous (Theorem

1.2.4.2 b)) and so v is also continuous. Take Z E E / F

and z C Z . There is a

pair (x,y) C F x G with

~(x, y) = z. It follows that vy = qy = q(x + y) = q ( u ( x , y ) ) = qz = Z .

Hence v is surjective. We show that v is lower bounded. Take y E G . Then

124

1. Banach Spaces

Ilyll ~< Ilxll + Ilyll = II(x, y)ll~ = Ilu-~(x + y)ll ~< Ilu-~ll IIx + ytl for every x E F (Proposition 1.2.1.4 a) ), so t h a t IlYll < Ilu-lil inf I]x + Yll - Ilu-lll IlqYl]

IvYll = IlqYll >

1

By Proposition 1.2.1.18 a), b), v is bijective and v -1 is continuous, i.e. v is an isomorphism. b =~ c is trivial. c :=> a. Let z E E . There is a y E G with qy = q z .

Hence z - y E F

and u(z - y,y) = z-

y + y = z,

i.e. u is surjective. We show t h a t u is lower bounded. Take (x, y) E F x G . Then

q(z + y) = qx + q~ = qy, so t h a t (Proposition 1.2.1.4 a), Theorem 1.2.4.2 b))

Ilyll =

v-~q(~ " + y)ll ~< IIv-'ll Ifq(x -4- y)fl ~< IIv-'ll I1~ + yll,

I1~11 = I x + y - y l l

~< I I ~ + y l l +

Ilyll ~< ( 1 + IIv-~lI)ll x + y l l ,

II(x, y)ll~ = sup{llxll, Ilyll) ~< (1 + IIv-~ll)llx + yll 1

II~(x, y)ll - I1~ + yll > - - I I ( x ,

y)ll

9

By Proposition 1.2.1.18 a), b), u is injective and u -~ is continuous. Since u is continuous (Proposition 1.2.5.1), it is an isomorphism. /

D e f i n i t i o n 1.2.5.3 orE.

[

0

)

Let E

be a n o r m e d space and F, G be subspaces

We say that E is the d i r e c t s u m o f F E=F|

if the m a p

1

\

a n d G , and we denote this by

1.2 Operators

F x G

>E ,

125

(z,y) ~ , '~ z + y

is an isomorphism. Let E be a normed space. A c o m p l e m e n t e d subspace of E is a subspace F of E for which there i s a s u b s p a c e G of E with E=F| In this case G is also a complemented subspace of E ; it is called a c o m p l e m e n t of F i n E .

By Proposition 1.2.5.2 a =~ b, complemented subspaces of a normed space are closed. But there are closed subspaces of Banach spaces which are not complemented subspaces (see Corollary 1.2.5.14). Let (E~)~ci be a finite family ofnormed space. For each A C I , the space E~ (canonically identified with a subspace of 1~ E~ ) is a complemented subspace of 1-I E~ 9

C o r o l l a r y 1.2.5.4 E , and G , H

Let E be a normed space, F a complemented subspace of

complements of F in E . Then G and H are isomorphic.

By Proposition 1.2.5.2 a =~ c, F is closed and G and H are isomorphic to E / F . Hence F and G are isomorphic to each other, C o r o l l a r y 1.2.5.5

m

Let E be a normed space and F be a complemented sub-

space of E . E is complete iff F and E / F

are both complete.

By Proposition 1.2.5.2 a =~ c, F is closed. Thus the assertion follows from Theorem 1.2.4.2 e),f), C o r o l l a r y 1.2.5.6

I

~

m

0

) Let F be a closed subspace of the normed space E

which is of finite codimension in E . Then E has a finite-dimensional subspace G such that E=FoG. As G we may take any algebraic complement of F in E . In particular, E is complete iff F is complete.

126

1. Banach Spaces

Let q : E -+ E / F for E l F .

be the quotient map and (XL)tE I a n algebraic basis

Given ~ E I , take x~ E X~ and let G be the vector subspace of E

generated by (x~)~Er. Then G is finite-dimensional and qIG is an algebraic isomorphism. By Corollary 1.2.4.9, qlG is an isomorphism, so by Proposition 1.2.5.2 c ::~ a E=FGG.

If H is an algebraic complement of F in E , then there is an isomorphism u : G ~ H . The map F•

>F•

(x,y),

>(x, uy)

is then an isomorphism, so E=F|

The last assertion follows from Proposition 1.1.5.1, since every finitedimensional normed space is complete (Corollary 1.1.3.5). [

D e f i n i t i o n 1.2.5.7

(

0

)

I

Let F be a vector subspace of the normed space

E . A projection in E is an operator p on E such that p o p = p. If F = I m p then we say p is a projection o f E onto F . In this case xEFc---~px=x for every x E E , so F is closed.

For every u E s

with u ( E ) C F , u is a projection of E on F iff

ux = x for every x E F .

Theorem

1.2.5.8 ( 0 )

(Murray,1937)Let F

and G be subspaces of the

space E . Then the following are equivalent:

~) E = F e a . b)

there is a projection p of E onto F such that G = K e r p .

If these assertions hold, then 1 - p is a projection of E onto G , F = Ker (1 - p), and E / F

is isomorphic to G.

1.2 Operators

a ==~ b. Let q 9 E -+ E / G

127

be the quotient map. By Proposition 1.2.5.2

a =~ c, G is closed and the map u " F ---+ E / G ,

x,

) qx

is an isomorphism. P u t j" F

) E,

p'-jou

x ~

>x ,

-1 o q .

Then p is an operator on E and I m p C F . Given x c F , p x = u - l (qx) = x .

Thus I m p = F and p o p = p. T h a t G = K e r p is obvious. b ==> a & the last assertion. Since (1-p) 1 -p

o(1-p)=l-p-p+p=l-p,

is a projection in E . Take x E E . T h e n (x e F ) ~

(x--px)

r

( ( 1 - p ) x = O) ~

(x e K e r ( 1 - p))

(x e G) r

(px = 0) ~

((1 - p ) x = z ) r

(x e Ira(1 - p)).

Thus F-Ker(1-p),

G-Im(1-p).

Define u "F x G v :E

~ E,

~ F x C,

( x , y ) ~-+ x + y , z ~

(pz, ( 1 - p ) z ) .

u and v are linear and continuous (Proposition 1.2.5.1, Proposition 1.1.5.1). Since uvz = pz + ( 1 - p ) z

= z,

for every z E E and v u ( x , y) = (p(x + y ) , (1 - p ) ( x

+ y)) = (x, y)

for every (x, y) C F • G , it follows t h a t u is an isomorphism. Hence E=F@G.

By Proposition 1.2.5.2 a =v c, E / F

and G are isomorphic.

I

128

1. Banach Spaces

C o r o l l a r y 1.2.5.9

Let E be a normed space, F a subspace of E ,

( 0 )

and G a subspace of F . a)

If F is a complemented subspace of E and G is a complemented subspace of F , then G is a complemented subspace of E .

b) If G is a complemented subspace of E , then it is a complemented subspace ofF. a) By Theorem 1.2.5.8 a =~ b, there are projections u of E onto F and v of F onto G. Then E

>

E,

x ~,

>

vux

is a projection of E onto G. By Theorem 1.2.5.8 b ::~ a, G is a complemented subspace of E . b) By Theorem 1.2.5.8 a ::v b, there is a projection u of E onto G. Then F

;,F,

X~

-)Ux

is a projection of F onto G. By Theorem 1.2.5.8 b =~ a, G is a complemented subspace of F .

II

Corollary 1.2.5.10

( 0 )

Let E , F

be normedspaces. Take u 6 s

such that the map E~Imu,

x,

) ux

is an isomorphism. Let G, H be vector subspaces of E and F respectively, such that G is not a complemented subspace of E . If u(G) c H and there is a projection of H onto u ( G ) , then H is not a complemented subspace of F . Assume H to be a complemented subspace of F . By Murray's Theorem,

u(G) is a complemented subspace of H . By Corollary 1.2.5.9 a), u(G) is a complemented subspace of F and by the same corollary b), it is a complemented subspace of Im u. It follows that G is a complemented subspace of E and this is a contradiction. E x a m p l e 1.2.5.11 of ~(T)

II

( 0 ) If T is an infinite set, then there exists a projection

o~ ~o(T) ~ h o ~ no~,~ i~ 2 and ~ y

~h

p ~ o j ~ t i o ~ ha~ a ~o~.~ at

least 2. co(T) is a complemented subspace of c(T) with codimension 1.

1.2 Operators

129

Let ~ denote {A C T I T \ A

is finite},

the filter on T consisting of the cofinite subsets of T . It is easy to see that c(T)

~ c(T),

x,

~x-

(lim x(~))eT

is a projection of c(T) onto co(T) with norm 2 and with one-dimensional kernel. By Murray's theorem, co(T) is a complemented subspace of c(T) with codimension 1. Let u be a projection of c(T) onto co(T). Set X

:=

?_re T .

Then, given t C T , et e co(T) ,

IleT-2etll=l,

I1~11 ~ II~(~T- 2e,)ll- IIx- 2e, II ~ Ix(t) -- 21 ~ 2 - Ix(t)l, and so Ilull _> l i m ( 2 - I x ( t ) l ) - 2. Remark.

m

This result will be generalized in Proposition 4.2.8.23.

Example 1.2.5.12 Let E be a normed space, T a topological space, S a closed set of T , and u:C(S,E)

~C(T,E)

an operator with (~)

IS = x

for every x E C ( S , E ) . Then {x C C ( T , E ) ] x = 0 on S} is a complemented subspace of C(T, E) . The map C(T, E)

~ C(T, E) ,

x.

~ x - u(xlS )

is a projection of C(T, E) onto {x C C(T, E) Ix = 0 on S} and the conclusion follows from Murray's Theorem. Remark.

m

Exercises 1.2.14, 1.2.16 (resp. Corollary 1.2.5.15) present examples

where such an operator u exists (resp. does not exist).

130

1. Banach Spaces

Example

1.2.5.13 ( 0 )

If a complemented subspace of go contains co,

then it is not separable. Let E be a complemented subspace of g~ containing Co and let u be a projection in go with E -

K e r u (Theorem 1.2.5.8 a =, b). Put F'-

Imu

and define

z~'F

)IK,

x,

)z(n)

for every n E IN. By Lemma 1.1.2.17, there is an uncountable set 91 of infinite subsets of IN such that A N B is finite for distinct A, B r 91. Let ff~ be a finite subset of 91. Given A c if3, put

C(A) "- A \ O B BE~3 B#A

and x "-- E

eC(A) "

AE~

Then I xll < 1. Since E contains Co, Ue A -- UeC( A )

for every A C 91. Hence U X --- ~

?-teA .

AE~

Take x' C F t . By the above,

ff3Eg3f (!21) AE!B

AE~

sup Ix'(ux)l ~ IIx'o ~ll ~ IIx'll Ilull xcE#

(Proposition 1.2.1.4 b), Corollary 1.2.1.5). Hence (x'(ueA))A~ is a summable family (Proposition 1.1.6.14 d =v a) and

{A e ~ I ~ ' ( ~ ) # O}

1.2 Operators

131

is a countable set (Corollary 1.1.6.7). Thus

:-- U

i

o}

nEIN

is also countable. Take A E 9.1\9./o. Then (~)(~)

= x'(~)

= o

for every n E IN, so that U e A = O.

Thus eA E Ker u -

E.

{CA I A E 92\9.10} is thus an uncountable subset of E such that

for distinct A, B E 9.1\920. Hence E is not separable. Remark.

I

Lindenstrauss (1967) proved that every infinite-dimensional com-

plemented subspace of t ~ is isomorphic to t ~ , and therefore not separable (Example 1.1.2.2). C o r o l l a r y 1.2.5.14

( 0 )

(Phillips, 1940) For no infinite set T are c(T)

and co(T) complemented subspaces of t ~ ( T ) .

Let S be a countable infinite subset of T. Given x E t ~ "x "T~

IK

'

t~

! x(t)

if t E S

/

if t E T \ S

0

Define 9e ~ ( s )

~e~(T),

x,

~.

Then t c~(S)

>Imu,

is an isometry, U(Co(S)) C co(T), and

x,

>ux

define

132

1. Banach Spaces

co(T)

>co(T),

y,

>esy

is a projection of co(T) onto U(Co(S)). By Example 1.1.2.3 and Example 1.2.5.13, Co(S) is not a complemented subspace of t ~ ( S ) , so that by Corollary 1.2.5.10, co(T) is not a complemented subspace of t ~ ( T ) . By Example 1.2.5.11, co(T) is a complemented subspace of c(T) so that by the above considerations and Corollary 1.2.5.9 a), c(T) is not a complemented subspace of ~ ( T ) . m C o r o l l a r y 1.2.5.15 Let T be an infinite set endowed with the discrete topology and let T* be its Stone-Cech compactification. Then there is no operator u "C(T*\T)

~ C(T*)

with the property that x = ux[(T*\T) for every x e C ( T * \ T ) .

By Example 1.2.5.12, the existence of such an operator would imply that {x C C ( T * ) I x - 0 on T * \ T } is a complemented subspace of C(T*), i.e. that co(T) is a complemented subspace of t ~ ( T ) , contradicting Corollary 1.2.5.14. m Proposition 1.2.5.16 Given x E ~ ( T )

Let T be a set and take p E [1,c)c]U{0}.

( 0 )

, define 5" ~P(T)

and for u c s

~ IK,

t,

Take x e g ~ ( T ) . Then 5 e s

p-t~~ b)

y,

>x y ,

define it" T

a)

> ~P(T),

~ (uet)(t). I]511- [Ixll~. We define

>s

x,

~ is linear and the map

g~(T) is an isometry.

>Imp,

x,

>~x

.~ 5.

1.2 Operators

c)

u E s

=~/~ E e ~ ( T ) ,

d)

9 ~ e~(T)~

~ =

e)

The map

Itall~ -< II~*ll-

~.

s is a projection of s f)

133

,~ s

u , ~ ~, u

onto I m p of norm 1.

I m p is a complemented subspace of E(lP(T)). a), b), c), and d) are easy to see. e) By c) and a) the map is a well-defined operator of norm at most 1. By

d), it is a projection of F_.(gP(T)) onto I m p . Hence it has norm 1. f) follows from e) and Murray's Theorem.

i

134

1. Banach Spaces

1.2.6 T h e T o p o l o g y of P o i n t w i s e C o n v e r g e n c e /

D e f i n i t i o n 1.2.6.1

(

0

\

)

Let S be a set, A a subset of S ,

space, and J~ a set of maps of S into T .

T a topological

.~A denotes the set 2c endowed with

the topology of pointwise convergence in A .

9rs denotes the set ~ endowed with the topology of pointwise convergence, i.e. with the topology on .%- induced by the product topology on T s . Let E , F be normed spaces. By Proposition 1.2.1.4 a), the topology of pointwise convergence on L:(E, F) is coarser than the norm topology of L:(E, F ) . In particular, the topology of E~ is coarser than the norm topology of E ' . /

P r o p o s i t i o n 1.2.6.2

(

0

) Let E be a vector space, F a vector space of

linear forms on E , and V' a O-neighbourhood in FE. Then there is a finite subset A of E such that

{x' C F i x

c A ~

Ix'(x)l_

1} C V'.

Given x' c F , there is an a E IR+ with x' c a V '

By the definition of the topology of pointwise convergence, there is a finite family (x~)~e, in E and a family (c~)Lei in IR+\{0} such that

{x' c E ' I ~ e I ~

I~'(x~)l _< <} c v',

The set

has the desired properties. Now we prove the last assertion. We may assume that A ~- 0. We put

c~ := 1 + sup Ix'(x)l e ]R+. xEA

Then, for each x c A,

Ii.'(m)l_< 1, C~ so that ~x' C V , i.e. x' E a V .

1.2 Operators

L e m m a 1.2.6.3

135

( 0 ) Let x' be a linear form on the vector space E , and

(Xtt)eCI a finite, nonempty family of linear forms on E with

N Ker x: C Ker x'. tEK

Then x' is a linear combination of the x: (5 9 I ) .

We may assume that the family (x~)~i is linearly independent. We prove the lemma by complete induction on the cardinality of I . Take A 9 I and put J := I\{A}. By the hypothesis of the induction, there is an

x~ ( n Kerx:)\Kerx~ ~EJ

(if J = q), replace the intersection by E). Put y' "=X'--

Xt

(X)

,

X'~ (X) x~ "

Take

rCJ

Then

(

)

, x~(y) x~ Y-x~(x) X =0, !

x~! (x) x C

NKer x~ '

o - x' (y) - ~'~ (y) x'(x) = y' (y)

~'~ (x)

N Ker x'~ C Ker y'. tEJ

By the hypothesis of the induction, y' is a linear combination of the x'~ (~ C J ) . Hence x' is a linear combination of the xt~ (5 C I). I L e m m a 1.2.6.4

( 0 ) Let A be a finite subset of the vector space E . Let

F be a vector space of linear forms on E and x" a linear form on F which is bounded on

136

1. Banach Spaces

{x' C F' l x c A ==a Ix'(x)l < 1}. Given x E E , define

Y" F

~ IK,

z'~

", z ' ( z ) .

Then x" is a linear combination of (X)xEA" In particular, there is an x E E with ~ -

x".

We may assume that A is not empty. Then N Ker2 C K e r x " , xcA I

and the assertion follows from Lemma 1.2.6.3. /

C o r o l l a r y 1.2.6.5

[

\

0

)

Let E

be a n o r m e d space, F

a vector subspace

of E ' , and x" a continuous linear f o r m on F E . Then there is an x C E such that

(x, .)IF -- x". By Proposition 1.2.6.2, there is a finite subset A of E such that x" is bounded on {x' E E ' l x

C A ==~ [z'(x)] _< 1}.

The assertion now follows from Lemma 1.2.6.4. P r o p o s i t i o n 1.2.6.6

I

( 0 ) I f E is a n o r m c d ,space, then the map

Ek

~+,

.~" *ll~'l

is lower semicontinuous and E '# is a closed set of E 'E .

Given x C E , the map

E~

~+,

~',

~x'(x)l

is continuous. The assertion now follows from Proposition 1.2.1.4 b). Remark.

E '# is even a compact set of E ~ . (See Theorem 1.2.8.1).

I

1.2 Operators

Proposition

1.2.6.7

137

If E is a normed space then the map

is continuous. Take (uo, vo) C s

# x s

A a finite subset of E , and c > 0. T h e r e

are neighbourhoods b/, 12 of uo and vo in s

and s

respectively,

such t h a t

II~o~- ~o~o~ I < ~ , for all u E b/, v C ]2, and x E A . It follows

IJ~v~- ~o~o~ll _< IJ~w- ~vo~li + I1~o~- ~o~o~ll < c

c

c

< l l ~ - ~o~ll + ~ < ~ + ~ = for all 'u E / g , v C 12, and x C A. Hence the above m a p is continuous,

i

138

1. Banach Spaces

1.2.7 C o n v e x Sets D e f i n i t i o n 1.2.7.1

( 0 )

The subset A of the vector space E is called convex (resp. absolutely convex) if a A + 13A c A

for every a, 3 E IR+ (resp. a, 3 E IK) with a

+)3 = 1

(resp.

I,~1 + I~1 _< 1).

It is easy to see that a subset A of a vector space is absolutely convex iff it is convex and aACA

for every a E IK with [a[ _< 1. P r o p o s i t i o n 1.2.7.2

( 0 ) 1] E is a normed space, then

{xCE[[[x[[
and { x E E l [ [ x [ [ <_o~}

are absolutely convex for every a E IR+.

Let x,y E E and a,/3 E IK with

I~1 + 191 < 1. Then IIax + ~/xlI -< lal IIxll + I~l Ily[I _< sup{iIxil, Ilyll} 9 The assertion now follows. P r o p o s i t i o n 1.2.7.3

I

( 0 ) Let E be avectorspace and A , B

be convex

(resp. absolutely convex) sets of E . Then a A + ~ B is a convex (resp. absolutely convex) set of E for every ol,)3 E IK.

Take 7, (~ E IR+ (resp. 7, 5 E IK) with 7+~=1

(resp. [7[ + [~[ _<1).

Then ,y(o~A + ~B) + 6(aA + 8 B ) = a ( T A + 5A) + ~(TB + 5B) C c~A + ~ B .

Hence a A + ~B is convex (resp. absolutey convex).

I

1.2 Operators

P r o p o s i t i o n 1.2.7.4

139

( 0 ) Let E be a vector space and (A~)~E, a nonempty

family of convex (resp. absolutely convex) sets of E . Then N A~ is convex

tEI

(resp. absolutely convex). Given any subset A of E , there is a smallest convex (resp. absolutely convex) set of E containing A ; it is equal to

{ E O~tXtI ((OQ,Xt))tEI tEI (resp.

finite family in IR+• A , E

tel

{ EC~LXLI ((OQ,XL))tEI finite family in IK x A, ~EI

c~ = 1 }

~ Ic~l <_ 1 } ) , tel

and it is called the convex (resp. absolutely convex) hull o$ A .

Take x, y E ['/A~ and c~,/3 E IR+ (resp. c~,/~ E IK) with tEI

c~+~=1

(resp. Icrl+l~l_< 1).

Then x,y E A~ and so ax + r

E A~

for every c E I . Hence c~x +/~y E N A~, i.e. ~ A~ is convex (resp. absolutely convex). LEI

The latter assertion is easy to see.

I

P r o p o s i t i o n 1.2.7.5 ( 0 ) The closure of a convex (resp. absolutely convex) set of a normed space is convex (resp. absolutely convex). Let E be a normed space and A a convex (resp. absolutely convex) set of E . Take x, y E A

and a , ~ E I R + c~ +/3 - 1

(resp. c~,~EIK) with (resp. I~1 + I~1 ~< 1).

There are sequences (Zn)nEr~, (Yn)nEIN in A converging to x and y, respectively. Then aXn + ~Yn E A for every n E IN, so that

c~x + / ~ y -

lim (c~xn +/~Yn) E A .

n---~oo

Hence A is convex (resp. absolutely convex).

I

140

1. Banach Spaces

C o r o l l a r y 1.2.7.6

( 0 ) Let B be the convex (rasp. absolutely convex) hull

of the subset A of the normed space E . Then B is the smallest convex (rasp. absolutely convex) closed set of E containing A , and it is called the convex (rasp. absolutely convex) closed hull of A .

The corollary follows immediately from Proposition 1.2.7.5. P r o p o s i t i o n 1.2.7.7

( 0 )

Let E , F

I

be vector spaces and u " E - 4

a linear map. Let A be a convex (rasp. absolutely convex) set of E a convex (rasp. absolutely convex) set of F .

Then u(A)

F

and B

is a convex (rasp.

absolutely convex) set of F and u I(B) is a convex (rasp. absolutely convex) set of E .

Take c~, ,3 E IR+ (rasp. c~,/3 C ]K) with c~+/3=1

(resp. Ic~1+1/31 <_ 1).

Then c~u(A) + flu(A) C u(c~A +/3A) C u ( A ) .

c~ul(B) + 3ul(B) C ul(c~B + fiB) C ul(B). Hence u(A) and ul(B) are convex (rasp. absolutely convex). P r o p o s i t i o n 1.2.7.8

1

( 0 ) Let E be a Banach space and ( X n ) n ~ a se-

quence in E converging to O. Then the convex (rasp. absolutely convex) closed

h~U of { ~

I ~ e ~)

i~ ~ompact.

Given x e g l (x(n)xn),~eiN is absolutely summable (and hence summable) in E (Corollary 1.1.6.10 a ==~ c)). Define u " ~'

~ E, x ,

~ ~-~ x(n)x,~. n c IN

u(e ~#) contains {x,~ I n C IN}. Since e ~# is absolutely convex (Proposition 1.2.7.2), u(f 1#) is also absolutely convex (Proposition 1.2.7.7). We show that u(g 1#) is compact. Let ~ be an ultrafilter on gl# Put x'IN

~IK,

n~

>limy(n). Y,~

Then x C gl# Take e > 0 9There is an m C IN with n E IN, n > m. Given n E INm, there is an An E ~ with

xnil < ~ for every

1.2 Operators

2m(llxnll

+

141

1)

whenever y E An. Then

I ~ - ~ y l l - II ~

(z(~) - y ( ~ ) ) ~ l l _<

nEIN

m

o(3

E

_< ~

I~(~) - v(~)l I1~11 +

~

n=l

C

I~(~) - y(~)l I1~11 < ~ + ~ -

n=m+l

for every y E ~ An (Corollary 1.1.6.10). Hence u(~) converges to u x , s o that n=l

u(g 1#) is compact. Let A be an absolutely convex closed set of E containing {xn I n E IN} and take x E gl#. Then m

rt--1

for every m E IN. Thus u x E A . Hence u(t! 1#) C A and u(g 1#) is the abso lutely convex closed hull of {xn I n E IN}. The convex closed hull of {Xn I n E IN} is a closed subset of u(g 1#) and is therefore compact, D e f i n i t i o n 1.2.7.9

m ( 0 )

Let E

be a vector space and A a convex set of

E . A face of A is a n o n e m p t y convex subset B of A with the property that

9,y c A, ~ E]0,1[, ~ x + ( 1 - - ~ ) y E

B~x,y

9 B.

A n extreme point of A is an element x of A such that {x} is a face of A .

A is, of course, a face of A and the face of a face of A is a face of A. In particular, if B is a face of A, then x E B is an extreme point of B iff it is an extreme point of A. P r o p o s i t i o n 1.2.7.10

( 0 )

Let E , F

be vector spaces and u " E --4 F a

linear map. Let d be a convex set of E and B a face of u ( A ) . Then A N u l ( B ) is a face of A .

By Proposition 1.2.7.4 and Proposition 1.2.7.7, u ( A ) and A N ul(B) are convex. Take x, y E A and a E ]0, 1[, with a x + (1 - a ) y E d N u l ( B ) .

1.~2

1. Banach Spaces

Then ux, uy E u(A) and

a u x + (1 - a ) u y

E B.

Hence ux, uy E B , i.e. x,y. E u l ( B ) . Whus A M u l ( B ) i s a f a c e o f Proposition

1.2.7.11

( 0

)

A.

I

If E ~ {0} is a normed space and x an

extreme point of E # , then [[x[[ = 1. If [[x[[ = O, then 1 1 x = ~y + ~(-y)

for every y E E # , so t h a t x cannot be an extreme point of E # . If 0 < IIxi] < 1 then 1 --x, I1~11

0EE #

IIxiI (~]x-~X) + ( 1 -

iixl) 0 - x ,

so t h a t x is not an extreme point of E # . Hence [Ixi] = 1.

Let T be a nonempty set. x E gl ( T ) # is an extreme point

Example

1.2.7.12

oS ~ I ( T ) #

i/y I~(t)l = 1 So~ ~om~ t E T

First assume that

I

Ix(t)l = 1 for some t E T . Take y , z E s

a E]0, 1[ with a y + (] - a ) z = x Then

ay(t) + (1 - a ) z ( t ) = x(t) so t h a t y(t) = z ( t ) = x ( t )

Since x, y, and z all vanish on T \ { t } ,

y=z=x. Thus x is an extreme point of ~I(T)#

and

1.2 Operators

Now a s s u m e t h a t

143

x is an e x t r e m e p o i n t of t!l(T) # . A s s u m e f u r t h e r t h a t

Ix(t)l # 1 for every t 9 T . By P r o p o s i t i o n 1.2.7.11, lixlll = 1. Hence t h e r e are :tistinct e l e m e n t s s, t 9 T with

9(~) # o,

x(t) # o

Define

x(s) y'T

> IK,

rl

>

if r - - S

(1 + Ix(s)l]

if r = t

0

if r = T \ { s , t } ,

0

z "T

> IK,

r'

~,

if r = s

( 1 + Ix(s)l Ix(t)l]~

x(t)

if r = t

x(r) T h e n y, z 9 g l ( T ) #

if r = T \ { s , t } .

and

Ix(~)l Ix(t)l Ix(~)I + Ix(t)l y + Ix(~)l + I~(t)I z = x. This c o n t r a d i c t s the h y p o t h e s i s t h a t x is an e x t r e m e p o i n t of t~l(T) # . Hence ix(t)l = 1 for s o m e t 9 T .

i

T h e above result a d m i t s a generalization. Example

1.2.7.13

Let # be a Radon measure on the Hausdorff space T .

Then x 9 L I ( # ) # is an extreme point of L I ( p ) # iff Ix(t)p({t})l - 1

for some

t 9 T.

Ix(t)#({t})l-

for s o m e

t 9 T.

First a s s u m e t h a t 1

Take y , z E L I ( # ) # a n d a e ] 0 , 1[ w i t h a y + (1 - a ) z - x . Then

a y ( t ) + (1 - a ) z ( t ) - x(t) , so t h a t

144

1. Banach Spaces

y(t)

- z(t) = ~(t)

Since x , y , and z all vanish #-a.e. on T \ ( t ) ,

y=z-x. Thus x is an extreme point of LI(#) # . Now assume that x is an extreme point of LI(#) #

Assume further t h a t

Ix(t)p({t})l # 1 for every t C T . By Proposition 1.2.7.11,

Ilxll

- 1. Hence there are two disjoint

compact sets K, L of T with oe

9- / ~ I~1dl,I # 0,

9 - f~ I*1 d #1 # O.

Define T y . _ c~ + / 3 x e T + xer\(KuL) ' o~

Z

"--

T

+ ~ x e T + XeT\(KuL).

Then y, z C L 1(#)# a::d c~ + / 3 y + a + /3z = x . This contradicts the hypothesis t h a t x is an e x t r e m e point of L 1(#)#. Hence

Ix(t)p({t})l- 1 for some t C T . Example

1.2.7.14

m

Let T be a Hausdorff space, .h4b(T) the Banach space

of bounded Radon measures on T (Example 1.1.2.26), and take p C .hdb(T) # Then # is an extreme point of .A4b(T) # iff there is a t C T such that Suppp-

{t},

Ip({t})l-

1.

M " - {~' C .h4b(T) ] ~, is positive and ~ ( T ) -

is a face of .A/Ib(T) # and

1}

1.2 Operators

145

{at It E T} is the set of extreme points of All, where at denotes the Dirac measure on T at the point t E T (i.e. at(A) =eA(t) for every Borel set A of T ). Assume t h a t

# is an extreme point .hdb(T) # . Assume further that the

support of # contains at least two points. T h e n there are disjoint Borel sets A and B of T such that A tO B : S u p p p ,

IpI(A)

--/: o,

IpI(B) :/: o.

Define 1 #A - - i P l ( A ) e A - # ,

1

#s -

I~I(B)eB

.#.

Then #A, PB E A//(T) # ,

I~I(A) + I#I(B) _< 1,

# =

I#I(A)~A + I#I(B)~B,

contradicting the hypothesis that p is an extreme point. Hence there is a t E T with Suppp={t},

I>({t})l=l

(Proposition 1.2.7.11). T h e converse is easy to see. The last assertion follows from the first one. I

( 0 ) Let T be a completely regular space. Then x E C(T) is an extreme point of C(T) # iff Ixl = eT. Example

1.2.7.15

Take x E C(T) # and assume t h a t there is a t E T with Ix(t)l r 1. Let U be a neighbourhood of t such that := sup I~(~)l < 1. sEU

Take y E C(T) with {y r O} C U , Then

[lYi[oo= 1 - a .

146

1. Banach Spaces

x+yEC(T)

l(x+y)

+

1

# ,

(x-y)-x

Thus x is not an extremal element of C(T) # . Now assume that Ix I = aT. Take y , z E C(T) # and a E ]0, 1[ with a y + (1 - a ) z = x . Then

ay(t) + (1 - a ) z ( t ) = x ( t ) , so t h a t ~(t) = z(t)

- ~(t)

for every t E T . Hence y~-z

--X

and x is thus an extreme point of C(T) # . Example e~(T)

1.2.7.16

Let T be a set. Then x E g~

I

is an extreme point oj"

~/y Ixl = aT.

This assertion follows immediately from Example 1.2.7.15. Example

1.2.7.17

I

Let T be a locally compact space. Then x of Co(T) is an

extreme point of Co(T) # iff x ] -

aT. In particular, Co(T) # admits extreme

points iff T is compact. Take first x E C0(T) # and assume t h a t there is a t E T with Ix(t)I # 1. Let K be a compact neighbourhood of T , so that a := sup Ix(~)l < 1. sEK

Take y E Co(T) # with {y#O}cK, Then

Iiyil~=l-a.

1.2 Operators

14 7

x :t= y C Co(T) # ,

l(x+y) +

1

(x-y)=x

Thus x is not an extreme point of Co(T) # . Now suppose that Ixl = eT. Then T is compact and, by Example 1.2.7.15, x is an extreme point of Co(T) # . C o r o l l a r y 1.2.7.18

9

Let T be a compact space and S a locally compact n o n -

compact space. Then C(T) and Co(S) are not isometric,

m

148

1. Banach Spaces

1.2.8 T h e A l a o g l u - B o u r b a k i T h e o r e m T h e o r e m 1.2.8.1

( 0 ) (Alaoglu 1940, Bourbaki 1938) Let E be a normed

space. Then E~E# (i.e. the unit ball of E' endowed with the topology of pointwise convergence) is compact. If E is separable then E~E# is metrizable. Let ~ be an ultrafilter on E '#

y':E

Define

>IK, x,

>limx'(x); x ! ,~

y' is linear and

ly'(x)l- lim Ix'(x)l ~ II~ll x I~ for every x e E (Proposition 1.2.1.4 a)). Hence y' is continuous and Ily'll ~ 1 (Proposition 1.2.1.1 d =~ a), i.e. y' E E '# . Since ~ converges to y' in the topology of pointwise convergence, E~ # is compact. Now suppose that E is separable. Let A be a countable dense set of E . Since E '# is equicontinuous, the topology of pointwise convergence in A coincides with the topology of pointwise convergence in E , i.e. E~ # - E~ # (Proposition 1.1.2.15). But E~ # is metrizable (since A is countable). Hence E~ # is metrizable.

1

T h e o r e m 1.2.8.2

( 0 )

(Banach-Dieudonn~)Let E be a normed space

and 92 the set of subsets A of E , such that

{x 9 A I Ilxll ~ c} is finite for every e > O. Then of the topologies on E ' , the topology of uniform convergence on the sets of 92 is the finest one to induce the topology of pointwise convergence on the equicontinuous sets of E ' . Put ~s

~ U I C E'

[

given an equicontinuous set A' of E ' , A' N U' is an open subset of A~

D

J

It is easy to see that T' is the finest tology on E' inducing the topology of pointwise convergence on the equicontinuous sets of E ' . Let |

be the topology

(on E ' ) of uniform convergence on the sets of 9.1. Since the sets of 9,1 are relatively compact, |

induces the topology of pointwise convergence on the

equicontinuous sets of E (Proposition 1.1.2.15). Hence |

C T'.

1.2 Operators

149

Take U' E ~s and x' E U'. Let n E IN. Given A c E , put

ft. "- {y' E n E ' # I x E A ~

I ( x , y ' - x'l] < 1}.

Then

n

is a downward directed set of closed sets of nE'#E, the intersection of which is {x'} n n E '#. Since U ' N n E '# is an open set of nE'#E and since nE'#E is compact (Theorem 1.2.8.1), there is an An E ~ I ( 1 E #) with

An C U'. Define A-

UAn nE IN

.

Then A E 91 and

x' E {y' E E' I x E A

~

I<~,y'-x'>l ~ 1} C U'.

Hence U' is a neighbourhood of x' with respect to | Thus G ' -

%'.

We deduce that ~' C | I

C o r o l l a r y 1.2.8.3

( 0 ) Let E be a Banach space and ~ the set of compact

convex sets of E . Then the topology on E' of uniform convergence on the sets in .~ is the finest one to induce the topology of pointwise convergence on the equicontinuous sets of E ' . Endow E' with |

the topology of uniform convergence on the sets of

and let ~s be the finest topology on E' inducing the topology of pointwise convergence on the equicontinuous sets of E' (Theorem 1.2.8.2). We have to prove that |

= ~'. |

C ~s follows from Proposition 1.1.2.15. By Theorem

1.2.8.2, to prove the reverse inclusion it suffices to show that, given a sequence (Xn)nEIN in E converging to 0, there is a K E ~. containing {Xn t n E IN}. But this was proved in Proposition 1.2.7.8.

I

150

1. Banach Spaces

1.2.9 B i l i n e a r M a p s Definition 1.2.9.1

( 0 ) Let E, F, G be vector spaces. A map u" E x F --+ G

is called bilinear if it is linear in each variable (i.e. u(x, .) and u(., y) are linear for every x E E and y E F ).

Proposition 1.2.9.2

(0 )

Let E, F, G be normed spaces and u" E x F--+ G

a bilinear map. Then the following are equivalent:

a) u is continuous. b) u is continuous at (0,0). c)

There is an a E IR+ such that

Ilu(~, y)il < -Ilxll Ilyll for every ( x , y ) E E

d)

x

F.

The restriction of u to E # • F # is uniformly continuous.

d~a~b

is trivial. b ==> c. There is a 5 > 0 so that liu(x,y)]l <_ 1

whenever (x, y) C E x F satisfies

li(~, y)I~ < 5. Put

1 OL

Take (x,y) E E x F

with x # 0 ,

"--

y~0.

= sup lj xl

~

52

.

Then

5

SO

52

5

5

<1

1.2 Operators

151

1

I1~(~, y)ll _< ~1 ~11 Ilyll = ~11~11 Ilyll c =~ d. Given (xl,yt), (x2, y2) C E # x F #,

Ilu(xl, y~) - u(x2, y2)ll < Ilu(x~, Yl)

-

-

~t(Xl, Y2)II + IIu(x,, Y~) -- U(X~,

_< IlU(Xl, Yl -- Y2)II-1-]]~t(Xl--< O~llXl]l

Ilyl

Ily=ll

x21l _< 2ll(x~, yl)

-

-

<

(x2,

Y2)II1 9

II

( 0 ) Given norm~d ~ p a ~ E, F, G,

Corollary 1.2.9.3 s

X2, Y2)II -~<

- yull + CtllXl -- X2[]

< ~lly~ - y=ll + ~llXl -

Y2)II --<

F) x s

G)

>s

G) ,

(u,v),

>vou

is bilinear and continuous.

It is easy to see that the map is bilinear. By Corollary 1.2.1.5 and Proposition 1.2.9.2 c ==v a, it is also continuous, m Corollary 1.2.9.4

( 0 ) If E, F are normed spaces, then the map Exs

>F ,

(x,u) ,

>ux

is bilinear and continuous.

The assertion follows from Proposition 1.2.1.4 a) and Proposition 1.2.9.2 c=>a. [] Corollary 1.2.9.5

( 0 ) Given a normed space E , E x E'

IK•

>IK,

~E,

(~, ~') ~-~ (~, ~') (~,x),

>~z

are bilinear and continuous,

Proposition 1.2.9.6

m

( 0 ) Let E , F , G

be Banach spaces, and u " E •

F ~ G a continuous bilinear map. Let (xn)ne~u{0}, (Yn)n~U{O} be absolutely summable families in E and F , respectively. Given n E IN, put zn := ~

U(Xk, Yn-k)

k--O

.for every n e IN U {0}. Then (zn)~c~u{o} is absolutely summable and

~u{o}

~e~u{o}

nc~u{o}

152

1. Banach Spaces

Given p c IN, define

Then limap=u(

E

x,~,

n~INu{o}

p---~ oo

E

Y~)

nc~u{o}

and (bp)pe~ is a convergent sequence. By Proposition 1.2.9.2 a =~ c, there is an a E IR+ such that

II~(x, y)ll _< o~llxll Ilyll for every (x,y) E E x F . Take e > 0. There is an rn E IN such that bp - b m <

E

l+a for every p E IN with p >_ m . Take p C IN, with p _> 2m and let A

:-- {(i,y)e (~,, u {o})x (~,, u {o})I,~ < sup{i,j}},

B 9=

,~(~'J)~ A l~ + ; >

~}.

Then p

(i,j)EB

n=O

-<

~ Z

(i,j)cB

II~ Ilyjl = ~ ( b , - b ~ ) <

~.

(i,j)EA

Thus

( zo-ap)-O, P

p--~oo

n--O

P

u(

E x,~, E Y'~) = l i m a p - p~oo n~u{o} ~e~u{o} n=o

Since 71

IIz~ll _< ~ ~

IIx~ll Ilyn-kll

k=0

for every n C IN, it follows that (zn)ne~u{o} is absolutely summable and that

x;

~e~u{o}

neINu{O}

~e~u{O}

1.2 Operators

153

Exercises E 1.2.1

Let E , F

be normed spaces and u " E - +

F be a linear m a p such

that, if (Xn)nE~ is a null sequence in E (i.e. lim x~ = 0), then (uxn)nc~ is n--+ (x)

bounded. Show that u is continuous. E 1.2.2 1)

sup

Let E, F be normed spaces and u" E --+ F a map, satisfying Iluxll <

oo.

xEE#

2)

z, y E E ~ u ( x + y) = ~

+ uy.

3)

If ]K = C , then u(ix) = iuxfor e v e r y x E E .

Show t h a t u is linear and continuous. E 1.2.3

Let E be a normed space. For x E E define

Ux" ]K

>E ,

a,

>a x .

Prove the following: a)

Given x E E , ux E s

b)

The m a p

E) and

E

~s

I1~11- Ilxll.

E),

z,

>u~

is an isometry. E 1.2.4

Let E , F

be normed spaces,

bounded sequence in s

A

a dense set of E , and (un)ne~ a

F) such that (UnX)nE~ is a Cauchy sequence (resp.

a null sequence) for each x E A. Show t h a t (UnX)nE~ is a Cauchy sequence (resp. a null sequence) for each x E E . E 1.2.5

Take p E [ 1 , c c [ , q E ] l , cc] with ~1 + q 1 - - 1. Show that (x)

lim /

(:x:)

x(s + t)y(t)dt - /

s--~ O -- oo

whenever x E s

y E f_.q(lR).

--oo

x(t)y(t)dt

154

1. Banach Spaces

E 1.2.6

Let E , F

be normed spaces, (x',y') E E' • F ' , and

z'-E•

~,

(~,y),

>~'(~)+~'(y)

Show that z' E (E • F ) ' and

llz, ll (ll ,ll § ii 'll )

1

where E x F is endowed with the 2-norm. E 1.2.7'

Let T be a set. Take x E f ~ ( T ) , and define x'el(T)

>]Z,

y,

>Ex(t)y(t) tET

Show that the following are equivalent (Example 1.2.2.3 b))" a)

There is a y E t~I(T) with [[Y[I = 1 and 5(y) - I ] ~ [ ] .

b)

There is a t 9 T with Ix(t)I = IIx ]oo.

E 1.2.8

Let T b e a s e t . Take x E t ~

Xco(T)

(resp. c(T) )

>IK,

Y'

>E x ( t ) y ( t ) " tET

Show that the following are equivalent (Example 1.2.2.3 b) ): a)

There is a y E co(T)

(resp. y E c(T)) with i l y l I - 1 and

I~(y)i = li~ll. b)

x E IK (T)

(resp. the map

T\~'(0)---+~,

~,

>

i~(~)l

is in c(S)). E 1.2.9

Take p,q, r E [ 1 , c r

with ~1 + q _1 > 1 and k" IN • IN --+ ]K such that

k(m, .) E g' for every m E IN. Given x E gq, define 9IN

~IK,

m,

~ Ek(m'n)xn" nE IN

Show that if ~ E fr whenever x E t~q, then the map X,

is continuous.

)X

1.2 Operators

E 1.2.10

155

Take k e C([0, 1] x [0, 1]). Given x 9 C([0, 1]), define 1

kx" [0,1]

> IK,

/ k(s, t)x(t)dt.

s,

0

Show the following: a) kx C C([0, 1]) whenever x e C([0, 1]). b) The map ,.,.,

k -c([o, 1])

x,

> C([0, 1]),

>kx

is linear and continuous and 1

Ilk I -

E 1.2.11

sup

~e[o,q

/

Ik(s, t)ldt.

0

Take p C [1, cx~]. Given x c l K 2z , define

Xr " 2Z

~ It(,

n,

> x( n-

l ),

xe" ~

> IK,

n,

> x(n + l).

Show that"

a) x e ~p(2z) ~ Xr, Xe e ~P(~), I x~llp --Ilxell~ --Ilxll~. b) x c c(2Z)

(resp. Co(2Z)) =:> xr,xe C c(~)

(resp. Co(2Z)).

c) The maps eP(~) --+ eP(~), c(2Z) -+ c(2Z), Co(2Z) -+ Co(2Z) defined by x ~-~ Zr (resp. xe) are isometries. They are called the right (resp. left) shift of t P ( ~ ) , E 1.2.12

Let

c(2~), and Co(~), respectively.

(OLn)nEiN

1 1 be a sequence in lK. Take p, q C ]1, c~[ with p+~ - 1,

and put

F'-{xcgPl

lirn ~-~akXk--O }. k--1

Show that the following are equivalent: ~3

a)

Y ~ l a n l q = O or c~. n=l

156

b)

1. Banach Spaces

F is dense in fP.

E 1.2.13

Let F be a closed vector subspace of the Banach space E ,

q : E --+ E l F

the quotient map, and T a set. Take u 9 ~_.(fl(T),E/F)

> 0. Show that there is a v 9

E) such that

u = q o v,

E 1.2.14

and

Ilvll ~< (1 + c)ll~ll,

Let T be a metrizable topological space, S a closed set of T , and

E a normed space. Prove the following: a)

There exists an operator

u:C(S,E)

>C(T,E)

of norm 1 such that (uz) I X = z

for every x 9 C(S, E) (see e.g. Jun-Iti Nagata, Modern general topology, Theorem VII.14). b)

{x 9 C(T, E) I x = 0 on S} is a complemented subspace of C(T, E ) .

E 1.2.15

Let E be a normed space, T a completely regular space, S a

closed set of T , A a countable subset of T such that A \ U is finite for every neighbourhood U of S and put f := {~ e

C(T, E)

Ix = 0 o. St.

Show t h a t there is an operator

U : co(A, E)

~ .T"

of norm 1 such that

(uz) i A = x for every x C co(A, E ) . E 1.2.16

Let E be a normed space, T a completely regular space, and K a

compact set of T . For each closed set S of T let S' denote the set of t c T such that t is not a point of adherence of S \ { t } , and define Kr for each ordinal number { by means of transfinite induction as follows:

1.2 Operators

157

Ko--K, nE~

Assume that there is a countable ordinal number ~ with K~ - O. Show that there is an operator

u" C(K, E)

~ C(T, E)

such that

(ux) I / ( -

x

for every x C C(K, E) and that {x E C(T, E) ] x = 0 on K} is a complemented subspace of C(T,E). (Hint" Use the preceding exercise.) E 1.2.17

Let E be a vector space,

A a convex (resp. absolutely convex)

set of E , (x~)~ci a finite nonempty family in A, and (a~)~i a family in IR+ (resp. in IK), such that Ea~-I

(resp.

~EI

Zla~]_~l). LCI

Prove the following:

a) ~a~x~CA. LCI

b)

There is an x C A such that E

E 1.2.18

x~ = (Card I) x.

Let E be a (separable) normed space. Show that there exists a

(metrizable) compact space T such that E is isometric to a subspace of C(T). E 1.2.19

Let E be a separable normed space. Show that E is isometric to

a subspace of t~~ E 1.2.20

Let E, F be normed space with F finite-dimensional. Show that

the closed unit ball in s

F) endowed with the topology of pointwise con-

vergence is compact (and metrizable if E is separable) (generalization of the Theorem of Alaoglu-Bourbaki).

158

1. Banach Spaces

E 1.2.21

Let E, F , G be normed spaces. Define

B(E, F; G):= {u " E x F

~ G lu is bilinear and continuous}

and

Ilull :-inf{o~ e JR+ [(x, y) e E x F ~ ~. E ~

~(F, C ) ,

9~

II~(x, Y)II ~ o~llxll Ilyll},

u(~, .)

for every u E B(E, F; G). Prove the following: a)

B(E, F; G) is a vector subspace of B(E,F; G)

G ExF and the map

~]R+,

u,

>llull

is a norm. b)

u is linear and continuous and

c)

The map

I1~1-lull for every u e B(E,F; G).

B(E,F;G)~E.(E,E.(F,G)),

u:

~u

is an isometry. E 1.2.22

Let F be a closed vector subspace of a Banach space. Show that

the set of projections of E on F is a convex set of / : ( E ) , closed with respect to the topology of pointwise convergence.

1.3 The Hahn-Banach Theorem

1.3 T h e H a h n - B a n a c h

159

Theorem

The H a h n - B a n a c h Theorem is the most i m p o r t a n t result in the theory of normed spaces, without which the theory would lose all interest. It ensures t h a t the dual space of a normed space contains sufficiently many vectors to allow every normed space to be isometrically imbedded - - by means of the evaluation map into its bidual. It also allows us to associate to each operator its transpose and bitranspose. The evaluation map enables us to define the most important class of Banach spaces: the reflexive ones.

1.3.1 T h e B a n a c h T h e o r e m Lemma

1.3.1.1

( 0 )

Let E be a vector space and F , G

vector subspaces

of E with F A G -- { O} . Put H:=F+G. Let x I and yt be linear forms on F and G , respectively. Then there is a unique linear f o r m z ~ on H with

z ' l F = x',

z'lG = y '

The uniqueness is obvious. Take (x~, Yl), (x2, y2) E F • G with x l + yl = x2 + y2. Then Xl-X2--y2-Yl

C FNG,

so t h a t Xl = x2,

Yl --- Y2 9

It follows that the map z':H

>IK,

x+y,

>x'(x)+y'(y)

is well-defined, z' has the required properties. Theorem

1.3.1.2

( 0 )

(Banach, 1929) Let F be a vector subspace of the

real vector space E . Let p be a real f u n c t i o n on E F such that

I

and y' be linear f o r m on

160

1. Banach Spaces

a)

p(x + y) < p(x) + p(y) for every x , y 9 E .

b)

p(ax) = ap(x) for every x 9 E and a 9 IR+.

c) y'(y) <_ p(y) fo~ ~ v ~ y y e F.

x'lF-

Then there is a linear form x' on E such that every x 9 E .

y' and x'(x) < p(x) for

Let ~ be the set of pairs (G, z') satisfying the following conditions: 1)

G is a vector subspace of E containing F ;

2)

z' is a linear form on G with z'lF = y';

3)

z'(z) < p(z) for every z 9 G .

Define an order on ~ by

(c,, z',) < (c~, z;)

<=~ 9 c , c c ~ , z~lC, - z,. !

!

It is easy to see that this order is inductive. By Zorn's L e m m a $2 has a maximal element (G, z'). We show that G = E . So suppose that G :~ E . Take x C E \ G and put H:={ax+ylaEIR,

ycG}.

Then H is a vector subspace of E , which strictly contains G. Then

~'(y) + z'(~) = z'(y + z) _< p(y + z) = p(y - ~ + ~ + ~) _< p(y - x) + p(~ + ~) for every y, z C G . It follows that

suv (z'(y) - p(y - ~)) < ~ ( p ( ~

+ ~) - z'(~)).

yCG

Take

Z e [~up (~'(y) - p(y - ~ ) ) , ycG

inf (p(z + x ) -

zcG

and let z~ be the linear form on H with

z~ta-

~',

z0(~) -

(Lemma 1.3.1.1). Take y C G and a > 0. Then

z'(z))]

1.3 The Hahn-Banach Theorem

161

~ o ( , ( 1 +~)-z'( 1 y)+ z,~(~y)) =~p(y+x)-p(~x+y), 1 ,

z0(-~

ly))<

+ y) - -~/~ + z'(y) = ~ ( - / ~

+ z'(~

z,(~xl + z,(l~l) : o~(1o

~_ o ( ~ ( ~ - ~1

xl

_

,( ox + ~

Hence (H,z~o) C ~ and (G, z') < (H, z;), m

which contradicts the maximality of (G, z'). E x a m p l e 1.3.1.3 on ~oo such that

Consider ] K -

lira inf -1 ~ n--+oo

IR. There is a continuous linear form x'

x,~<x '

(x) < _ l i m s u p -1~-~ xm n--~cxD n m--1

n m--1

for every x C goo. Then

I x'l

- 1,

x'lc0 - o,

x E t~~ ===>lim inf z~ < z'(x) < lim sup xn. n---+ o o

n--+oo

Moreover, XI 0 Ur

--

X t 0 72~ ~

X!

for every such x ' , where Ur (resp. u~ ) denotes the right (resp. left) shift on

Define p g~ 9

>IR,

x,

>limsup n--+ oo

--

T~

.~n ~_~ X

m

m--1

converges

X m

n m--1

1 _

nCIN

,

162

1. Banach Spaces

y'

F

9

>IR,

x~

> lim ~1 n-+cr

Xm

.

7Z

rn=l

Then F is a vector subspace of g ~ ,

y' is a linear form on F with y' < P l F ,

and

p(x + y) < p(x) + p(y) ,

p(.z) - .p(~)

for every x, y C gor and c~ E 1R+. By Banach's Theorem, there is a linear form x' on g~ with

x ' l F = y' ,

x' < p.

Thus

x' (x)

-x' (-x) > -p(-x)

__

__

lim inf ~1 n-+ cr

Xm

?'t

m=l

for every x E g ~ . The inqualities lim infxn _< lim inf -1 ~ n--+oo

n---+ cx~

T~

xm < x'(x) < lim s u p -1 ~ n - + cx:~

rn=l

rt

xm < lim sup xn n--+oo

m--1

are trivial for x C g ~ . We deduce that x' is continuous with norm 1 and vanishes on co. Take x E g~ and put y := utx

(resp. y := urx).

Then

1~ n

(xm

Ym) -- xn

(resp.

X 1 --

m=l

Xn+

1 ]

n

n

for every n C IN. Hence lim

-l~(xm-ym)-0.

n - + o o ?Z

m=l

Thus x - y E F

and * ' ( * - v) = v ' ( * - v) - 0 ,

Remark.

.'(x)

= x'(v),

tt

Let ~ be a free ultrafilter on IN. Then x"g~

x~

~liml n , 8 ~ rZ

Xrn m--1

has the properties required in the above example. This is another proof for the existence of x' which does not use Banach's Theorem. Of course, not all x' are of the above form.

1.3 The Hahn-Banach Theorem

P r o p o s i t i o n 1.3.1.4

( 0 ) Let E be a vector space and A a convex (resp.

absolutely convex) set of E with x E a A .

163

such that f o r every x C E

there is an a E JR+

Define

>IR, x ,

p:E

>inf{aEIR+ixcc~A}.

Then

p(x + y) < p(~) + p(y), f o r every x, y E E

p(.x) = I-Ip(x)

and (~ E ]R+ (resp. a C IK ).

Take /3, 3' e IR+\{0} with x C fl A ,

y c "),A .

1 1, Then ?x, ~y E A, so that

1

~(x fl+7

+ y) =

fl 1 3/ 1 -yEA. ~f l +-T f l x + fl + 7,, /

Hence x + y E (fl + 7 ) A ,

p ( x + y) < fl +'),.

Since fl and -), are arbitrary,

p(x + y) < p(x) + p(y). The other assertion is trivial. L e m m a 1.3.1.5

( 0 ) Let E , F

I be complex vector spaces and u

E -+ F

an I R - l i n e a r map. T h e n E

>f,

>ux-iu(ix)

x,

is C-linear.

Define v :E

) F,

x ,

>ux-

iu(ix).

v is obviously IR-linear and v(ix) = ~(i~) -i~(-~)

= ~(~

- i~(ix)) = ~x

for every x C E . Hence v ( ( ~ + i g ) x ) = v ( ~ x ) + v ( i g x ) = ~ v x + i f l v x = (~ + i g ) v x

for every x E E and a,/3 C IR, i.e. v is C-linear.

I

I 63

1. Banach Spaces

P r o p o s i t i o n 1.3.1.6 space E .

Let B

( 0 ) Let A be a n o n e m p t y convex set of the vector

be an absolutely convex set of E

there is an ~ C IR+ with y E a B .

such that f o r each y C E

Take x C E \ ( A + B ) .

Then there is a linear

f o r m x' on E , bounded on B , such that

sup re x' (y) < re x'(x). yCA

Now 0 E B. We may assume that 0 c A (otherwise we replace x and A by x - a

and A - a

for an a c A ) . W e p u t 1

C:=A+-~B,

p:E

~IR,

y,

~inf{a~IR+lyeaC},

y' : F .... ~ IR,

c~x ~-+ c~p(x) .

By Proposition 1.2.7.3, C is convex and by Proposition 1.3.1.4,

p(y + ~) < p(y) + p(~),

p(~v) = ~p(y)

for every y, z C IR+ and c~ E IR+. Take c~, fl E IR+ with

x e ( ~ B ) n (ZC). Then c~ > 1, fl > 1. There is a pair (a, b) E A x B, with 1

1

~-

a + ~b.

Then x-

( (11- ~)

x+

1-

(

b

1

=~x-

2 -aeA,

1

x+~bCB,

~) 1B B . 1-~ z~ 2 ' x~ ~2(/3-1)

1.3 The Hahn-Banach Theorem

165

2a 2a-

2 ( / 3 - 1)

1

Since x ~ C , it follows that 2cg p(x) > -2a-1

> 1.

Thus we have y' < p F . By Theorem 1.3.1.2, there is an N - l i n e a r map z " E ~ ]R such that z' < p and z' (x) = y'(=) = v ( * ) > 1.

Put X I .--- Z I

if I K - I R

and x' . E - - + r

v.

", z' (v) - iz' (iy)

if IK - C .

By Lemma 1.3.1.5, x' is a linear form on E . Since p _< 2 on B and

-B-B,

x' is bounded on B . T h e n s u p r e x ' ( y ) = supz'(y) < supp(y') <_ 1 < z'(x) = r e x ' ( x ) . yEA

Corollary

yEA

1.a.1.7

i

[

0

I

yEA

\

)

Let E be a norrned space, A a nonempty closed

(absolutely) convez set of E , and take x E E \ A .

Then there is an z' E E'

such that

sup re x' (y) < re x' (x), yEA

(sup Ix' (y)[ < x'(x)). yEA

Take c > 0 with V f (x) c E \ A .

Then x E E \ ( A + U E ( 0 ) ) . By Proposition 1.3.1.6 (and Proposition 1.2.7.2), there is a linear form x' on E , bounded on UE(0), such that sup re x'(y) < re x'(x). yEA

By Proposition 1.2.1.1 e => a, x' E E ' .

166

1. Banach Spaces

Now suppose that A is absolutely convex. Then x'(x) :/= O. Define

Y "

~'(~) ~,. Ix'(x)l

Take y 9 A with y'(y) :/= O. Then

x,(z) y,(y) Ix'(x)l ly'(y)l so that

ly'(y)l

(x,

y 9 A,

(x) y'(y) Ix'(x)l ly'(y)l y

= rex'

)

x'(z)

<- suprezeA x'(x)

s u p ly'(y)l < s u p r e x ' ( z ) < ~ - :~A

P r o p o s i t i o n 1.3.1.8

rex'(x)<

-

Ix'(x)l =

x'(z)l

'

x'(x) = y'(x).

I

Let E be a vector space, F a vector space of

( 0 )

linear forms on E , A' a nonempty (absolutely) convex closed set of FE, and take x' 9 F \ A ' .

Then there is an x 9 E such that sup r e y ' ( x ) < r e x ' ( x ) , yl E A

sup ly'(x)l < x ' ( x ) ) . y' 6A'

There is a 0-neighbourhood U' in FE with

x' + U' c F \ A ' . By Proposition 1.2.6.2, there is a finite subset A of E such that V' := {x' e Fix 9 A ~

]x'(x)l < 1} C U'

and for each y' 9 F there is an c~ 9 IR+ with y' 9 c~V'.

V' is absolutely

convex and x' 9 F \ ( A ' +

V').

By Proposition 1.3.1.6, there is a linear form x" on F , bounded on V', such that sup rex"(y') < r e x " ( x ' ) . y'EA'

1.3 The Hahn-Banach Theorem

167

By L e m m a 1.2.6.4, there is an x C E with

y'(.) = x"(y') for every y' E F . Then sup r e y ' ( x ) < r e x ' ( x ) . y' E At

Assume now A' absolutely convex. Then x'(x) ~ O. We set

~'(~) Y-

I~'(x) l

X.

Let y ' C A' with y ' ( y ) ~ O. T h e n

x'(x) y'(y) y, C A' Ix'(~)l ly'(z)l SO

sup re z'(x) >_ re zt C A'

x'(x) y'(y) y'(x)Iz'(x)l ly'(y)l

-- re ly'(y)l Y'(Y) y ' ( y ) -

ly'(y)l.

Hence sup lY'(Y)]-< sup z'(x) < r e x ' ( x ) < ]x'(x)l = x ' ( y ) . y; E A'

Corollary 1.3.1.9

I

z' E A'

( 0 )

Let E be a normed space and At an absolutely

convex closed set of E'E such that

llzll = ~up I~'(~)l xtEA t

]or every x C E . Then A ' -

E '#

Let x' C E ' \ A ' . By Proposition 1.3.1.8, there is an x E E such t h a t

~up ly'(~)l < x'(x). y' c A '

We get

llxll < ~'(x) < IIx'll ll~ll, so x' C E # . Hence E '# c A'. The converse inclusion is trivial.

I

168

1. Banach Spaces

Theorem

1.3.1.10

( 0 )

( K r e i n - V i l m a n ) L e t E be a normed space, K '

a convex compact set of E'E, L' the set of extreme points of K ' , and 92' the set of closed faces of K'E. The intersection of every nonempty family in 92' belongs to 9.1' if it is

a)

nonempty. b)

Every A' E 92' contains an extreme point of K ' .

c)

K' is the smallest closed convex set of E'E containing L'. a) Let (d't)~Ei be a nonempty family in 92'. Take x, y E K ' and c~ E ]0, 1[

with c~x + ( 1 - c ~ ) y E N A : " tEI

Then x , y E A[ for every t E / , s o that ["1 A[ is a closed face of K~ (Propotel

sition 1.2.7.4). b) Order 92' by reverse inclusion. By a) and Zorn's Lemma, there is a maximal clement B' of 92' contained in A'. Take x E E . The set

{*'(*)1-' E B'} is a convex compact set of IK (Proposition 1.2.7.7), and so it contains an extreme point c~. By Proposition 1.2.7.10,

{.' E B'I*'(*) = ~} is a face of B' and hence a face of K ' . It is obviously closed in E ~ , so that it belongs to 92'. Since B' is minimal, it coincides with B ' . Thus

. ' ( . ) = y'(~) for every x', y' E B ' . Since x is arbitrary, B' must be a one point set, and this point is our extreme point of K ' . c) Let M' be a convex closed set of E~ containing L'. Assume there is an x' E K ' \ M ' . By Proposition 1.3.1.8, there is an x E E such that sup rey'(x) < r e x ' ( x ) . y'EM'

The set

{y'(~)lv' E K'}

1.3 The Hahn-Banach Theorem

169

is a convex compact set of IK (Proposition 1.2.7.7), so that it contains an extreme point c~ with sup re y ' ( x ) < re c~. y'EM'

By Proposition 1.2.7.10,

{y' e K'Iv'(z) = ~} is a face of K ' , which is obviously closed in E ~ . By b) it contains an extreme point of K ' . This extreme point does not belong to M ' , which is a contradiction. Hence K ' C M' and K ' is the smallest convex closed set of E~ containing L'

I /

D e f i n i t i o n 1.3.1.11

\

( 0 ) A normed space E is called a dual space if there

is a Banach space F such that E is isometric to F ' .

F is called a predual

oSE.

Every dual space is complete (Corollary 1.2.1.10, Proposition 1.2.1.14). The notion "dual space" is a geometric and not a topological one. More precisely, there are isomorphic Banach spaces such that the one is a dual space whereas the other is not (see Example 1.3.1.16 c)). The predual of a Banach space is not unique. Indeed, Co x (g~/co) is a predual of (goo), (Corollary 1.2.2.14), but co x (g~/co) is not isomorphic to go~ (Remark of Example 1.2.5.13). C o r o l l a r y 1.3.1.12

( 0 ) If E is a dual space then E # has extreme points.

Let F be a Banach space and u : F' -+ E an isometry. By Proposition 1.2.7.2 and by the Alaoglu-Bourbaki Theorem, F '# is convex and compact. By the Krein-Milman Theorem, it has extreme points. It follows that E # also has extreme points. Example 1.3.1.13

I If T is a locally compact non-compact space, then Co(T)

is not a dual space.

By Example 1.2.7.17, Co(T) # has no extreme points. Hence, by Corollary 1.3.1.12, Co(T) is not a dual space. E x a m p l e 1.3.1.14

I

If T is a set, then co(T) is a dual space iff T is finite.

170

1. Banach Spaces

If T is finite, then co(T) is isometric to that co(T) is a dual space. If r

O(T)' (Example 1.2.2.3 g)), so

is infinite, then co(T) is not a dual space by

Example 1.3.1.13.

II

E x a m p l e 1.3.1.15

Let T

be a Hausdorff space. If # ~ 0 is an atomless

Radon measure on T , then L l(p) is not a dual space. By Example 1.2.7.13, LI(#) # has no extreme points, so that by Corollary 1.3.1.12, L!(#) is not a dual space. E x a m p l e 1.3.1.16

Let ~ be a free ultrafilter on IN and take

p . e~

a)

II

>/R+,

x,

>llxll~+limlx(n)l. n,~

p is a norm and

I1.11~ ~ p ~ b)

{x 9 e~lp(x)

211. II~.

<_ 1} has no extreme points.

~

endowed with the norm p is not a dual space but it is isomorphic to

~

endowed with the usual norm I1" Iior which is a dual space.

a) is easy to see. b) Let x be an extreme point of {x C e~lp(x) <_ 1} and take n E I N . Assume that ]x(n)l ~ 1. Define

y+ " IN

)IN,

k,

~ ! x(k)

(

if k=/:n

x(n) -t-- (1 - I x ( n ) l )

if k - n.

Then p(y+) _< 1 and 1

1

This contradicts the assumption that x is an extreme point of {x C f~lp(x) <_ 1}. Hence Ix(n)l = 1 for every n C IN. This leads to the contradiction that

p(~) = 2. c) By b) and Corollary 1.3.1.12, go~ endowed with the norm p is not a dual space. By a), it is isomorphic with g~ endowed with the usual norm I1" I1~, which is the dual of 61

9

1.3 The Hahn-Banach Theorem

171

1.3.2 E x a m p l e s in M e a s u r e T h e o r y (S. Banach, 1923) L e m m a 1.3.2.1

Let IK = IR. Let T be an additive group, f2 the class of all

finite nonempty families in T , and define

~" [2

) IN,

(t,)~EI'

> Card/.

Put utx " T

) IR,

s,

) x(s + t) ,

~ ]R,

s,

~ x(-s),

ux" T pw(X)"

=

1

sup 2~(~d)seT

( E u t L ( x + u x ) ) (s) tEI

for every x 9 e ~ ( T ) , t 9 T , and w := (t~)~E, 9 f2. Define further p(x) := inf p~(x) wE.Q for every x e g ~ ( T ) . Then for every x , y C g ~ ( T ) , a 9 IR+ , and t 9 T "

a)

infx(t) < p(x) <_ s u p x ( t ) . tET

tET

b) p(x + y) < p(~)+ p(y)

d) p ( x - utx) < O. ~) p(x - ~x) <_ o

f)

If T is a compact additive group and ~ is its Haar measure normed by )~(T) = 1, then

f ~dA <_p(~). g) If T is a topological additive group with the Baire property, then

)~EL

Ac L

for every finite family (a~)~EL in IR and every family (A~)~EL of dense Cs-sets of T .

172

1. Banach Spaces

a) is obvious.

b) T~ke ~,1 := (~,)~, e S~ ~nd ~ := ( t ~ ) ~

e S~. Put

w :-- (SL + t~)(~,a)elxL E ~Q. Then

1

p(x + y) < p~(x + y) =

(

sup

2~(~d) s e T

1

(

LEI

1 <- ~ ( ~ )

1 ~(~)

~,,+~(~ + y + ~x + w ) ) (~) -

( Z u ~ , ( x + ~ x ) ( ~)+ a ) +

suv ~

2~(w) ~eT

E

(t,,k)elx L

~L

~eI

AEL

1

sup

(

2~(~,) ~

1

sup 2~(~) ~,

EUs'(X+UX)

)

,~,

(

Eut~(y+uy) ~,~L

)

(s+t~)+

(s+s~)=

1

1

~(W2) ~ P ~

(x) + ~(~1) Z P~ (Y) = P~, (x) + p~ (v). tEI

Since czl and w2 are arbitrary,

p(x + v) ___p(~) + p(v).

c)

is trivial.

d) Take n C IN. Put w := (kt)kE~. C ~2. Then p ( ~ - ~ , ~ ) <_ p ~ ( x - ~ , ~ ) =

1

sup 2n set

x - utx + u x - uutx)(s + kt)

)=

1.3 The Hahn-Banach Theorem

173

n

~(

x(s + kt) - x(s + (k + 1)t))+ n k~l(X(-X= -- kt) - x ( - s - (k - 1)t))

1 sup 2n s e T

1

= - - s u p ( z ( s + t) - x(s + (n + 1)t) + x ( - s - nt) - z ( - s ) ) <_ 2ft sET

Since n is arbitrary,

p ( z - u~z) <_ O. e) p ( x - ux) i 51 sup(x - ux + ux - uux)(s) - O. seT

f) Take w "-(tt)tei e s =

Pw(S)

1 2~(W)

sup

(

sET

Then

Ettt'(x+ux) tEI

1 _> 299(w) ~

(

)

1 /(

(s) > 2qo(w)

/ / ) / ut~xdA -4- ut~uxdA ,

)

Eut'(x+ux)

,

tel

-

9

dA>

xdA. *

g) Take w "- (t~)tEi C ,(2. The sets - t t + Aa are dense Gs-sets of T for every (~, A) C I x L. Since T has the Baire property, we can choose

(N tel

+

(N tel

AEL

AEL

Then p~(Ea;~eAx ) =

1

sup(Eut~(Ea),(eAx

2(#(W)

)~EL

1

sup

ate(W)

1

sET

(

seT

E

Eax

tel

ACL

(

teI

+ U e A x ) ) ) ( s ) --

AEL

(

e-t~+A~ + e-t~-A~

)

1

Since w is arbitrary,

(

))

(s) >_

)

I AEL

ACL

174

1. Banach Spaces

E x a m p l e 1.3.2.2

Using the notation of L e m m a 1.3.2.1, let .T be a vector

subspace of e~176 and y' a linear f o r m on .7c with y' < p l Y .

Then there is

an x' 9 t~176 ' such that

1) IIx'll ~< 1; 2)

x' o ut - x' o u = x' for all t 9 T ;

3)

x'(x) 9 IR+ for every positive function x of e~176 ;

4) ~ ' l T = y ' . By Lemma 1.3.2.1 b), c), and the Banach Theorem (Theorem 1.3.1.2), there is a linear extention x' of y' on t?~(T) such that x' < p. Then

x'(x) ~ p(x)~ Ilxll,

- x ' ( x ) = x'(-x) ~ p(-x) ~ II- xll = Ilxll

(Lemma 1.3.2.1 a)). Hence

I~'(x)l < I1~11 for every x 9 ~?~176 Therefore x' is continuous and I1~'11 <_ 1 (Proposition 1.2.1.1 d ~ a). Take t 9 T and x 9 b~176 Then X t ( X - UtX) ~ p ( x -

UtX ) < O,

X t ( X - UX) ~ p ( x -

ux) < 0

(Lemma 1.3.2.1 d), e)). Since x is arbitrary, -x'(z

- ~)

= ~'((-x)

- ~(-~))

<_ o ,

-~'(~ - ~x) - x'((-x)

0,

Xt(X) -- X' o IZ(X) -- Xt(X

- ~(-.))

< o.

Hence Xt(X) -- X' o lZt(X ) -- X'(X -- lZtX )

=

-- /ZX) =

0,

and X ~ o~t

~

X~ o~/, =

X ~.

Thus

-~'(~) - ~'(-~) < p ( - z ) < 0 (Lemma 1.3.2.1 a)), so that

x'(z) >_ o for every positive function x of g~176

I

1.3 The Hahn-Banach Theorem

L e m m a 1.3.2.3

Let S be an infinite subgroup of the additive group T , ~

ring of subsets of T . Take C 9 ~ on ~

175

a

and let # be a bounded positive real function

such that for every A, B 9 9~ with A M B = O and every s 9 S :

1) #(AU B)

=

2) s + A 9 1 6 2

#(A)

+

#(B),

#(s+A)=p(A),

3) V(c) c ~ , 4) for every t c T there is an r c C with t - r r S . Then there is a subset B of C such that (s + B)ses is a partition of T and

,(~ + B ) = o for every s C S .

We define s ~ t : v=:~ s -

t c S

for s, t C T. ~ is an equivalence relation in T. By 4), there is a subset B of C containing exactly one point in each equivalence class of ~ . Then (s + B)scs is a partition of T. By 2) and 3), s + B E g~,

#(s + B) = p ( B )

for every s E S. Thus, by 1), #(B) CardD

=

Ep(s

+ B)=

sED

#( U(s

+

sED

B)) _< sup p(A) AC~

for every finite subset D of S. Hence #(s+B)=p(B)=O

I

for every s C S. Lemma 1.3.2.4

Let T be an infinite additive group and p : ~ ( T ) -+ JR+ a

map such that # ( A U B) = It(A) + It(B) for disjoint A, B C ~3(T) and

1"[6

1. Banach Spaces

#(t + A) = #(A) for every A 9 g3(T) and t 9 T . Then there is a partition (An)ne~ of T , such that

#(An) = 0 for every n 9 IN.

Let S be a countable infinite subgroup of T . The assertion follows from Lemma 1.3.2.3 by setting 9l : = g l ( T ) , E x a m p l e 1.3.2.5

Let T

C := T .

be a compact additive group and )~ be its Haar

measure normed by )~(T) = 1. Given x 9 ~ ( T ) utx : T

) IK,

ux : T Then there is an x' C g ~

1)

I

~ IK,

s,

and t 9 T , define

~ x ( s + t) ,

s,

>z ( - s ) .

with the following properties:

IIx'll - 1,

2) x' out = x' o u = x' for every t E T , 3) x'(x) r IF[+ for every positive real function in e~ 4) x'(x) = f xd)~ for every x E s176 If T is infinite, then for every such x' there is a partition (An)ne~ of T such that

9' ( ~ )

=o

for every n E IN. (Hence the Lebesgue Convergence Theorem does not hold for such an x' .)

First consider the case IK = IR. Let p be the function defined in Lemma 1.3.2.1. By Lemma 1.3.2.1 f),

I

x d~ <_ p(z)

1.3 The Hahn-Banach Theorem

for every x E s

177

At~~176 and the existence of x' with the desired properties

follows from Example 1.3.2.2. If IK - r then gOO(T)

>r

x,

)x'(rex)+ix'(imx)

has the required properties. The final assertion follows from Lemma 1.3.2.4. E x a m p l e 1.3.2.6

I

Let T be a topological additive group with the Baire pro-

perty. Given x c g~162 and t 6 T , define utx : T

) IK,

s,

ux : T

) IK,

s,

Then there is an x' E g~162

> x(s + t),

~ x(-s).

with the following properties:

1) IIx'll- 1, 2) x' o u t = x' o u = x' for every t c T , 3) x'(x) e

IR+ for every

positive real function x in g ~ ( T ) ,

4) x'(em) = 1 for every dense G~-set A of T . If T is infinite, then for every such x' there is a partition (A,~)n~ of T such that

9'(~o) = o for every n C IN. (Hence the Lebesgue Convergence Theorem does not hold for such an x' .)

First consider the case IK = JR. Let p be the function defined in Lemma 1.3.2.1, 91 the set of dense Gs-sets of T, ~ the vector subspace of g~176 generated by {eA I A C 92}, and y'" ~

>JR,

E ~ A691

'

(In the above sums,

{A ~ ~ I o~,4--/:0}

> E OZA" A691

178

1. Banach Spaces

is obviously finite and MA:~0 AE~I ~A~0

so the value of y' does not depend on the representation.) By the last assertion of Lemma 1.3.2.1, y' < p I ~ - The existence of x' with the desired properties now follows from Example 1.3.2.2. If IK = ~ , then /~~

~r

x,

) x'(rex)+ix'(imx)

has the required properties. The final assertion follows from Lemma 1.3.2.4. E x a m p l e 1.3.2.7

Let n 9 IN, ~ the Lebesgue measure on lR n, and

.7" := {x 9 g~(]Rn) I {x :/- 0}

is bounded}.

Define u t x : IR n

) IK,

ux : lR ~

s,

) IK,

) x ( s + t) ,

s,

~ x(-s)

for x 9 .T and t 9 1Rn . Then .T is a vector subspace of f~(]l=['~), u t x , ux 9 .T for every x 9 .T and t 9 IR n, and there is a linear f o r m x' on .T with the following properties:

1) x' o u t = x' o u = x' for every t 9 T , 2)

x'(x) 9 IR+ for every positive real function x in 3c,

We may replace 3) by

3')

There is a A-null set A with x'(eA) = 1.

For every such x' there is a dosjoint sequence (An)nelN Of subsets of ]R '~ such that

U An is bounded, n c IN

X'(en A ) 0

1.3 The Hahn-Banach Theorem

179

and

z'(~o) = 0 for every n E IN. (Hence the Lebesgue Convergence T h e o r e m does not hold for such an x' .)

Let T be the compact additive group IR~/2Z n , ~o : ]R ~ --4 T the quotient map, and x' the linear form from Example 1.3.2.5 (resp. Example 1.3.2.6). For every p

=

(Pk)kelNn E 2Zn the map n

I-[[pk,1 + pk[

>T ,

t ,

) ~o(t)

k=l

is bijective. Let ~op denote its inverse. The m a p

7

~IK, z, ~ ~

~'(xo~,)

pCTZ n

has the required properties (for 3', it is sufficient to take as A a Gs-set of ]R ~ which is dense in [0, 1[~ and which is a A-null set). In order to prove the last assertion, put

s : = Q ~, := {A C #:~

]Pu n I

> IR+,

c = [ 0 , 1 [ ~, A is bounded }, A ~ " x'(eA).

By L e m m a 1.3.2.3, there is a B C C such t h a t (s + B ) , e s is a partition of T and

~(~ + B ) = 0 for every s c S . Given s c S , we put As:=(s+B)

A [ 0 , 1 [ ~.

Then x'(As) = 0 for every s C S and

X'(enAn) x'(eC01E)

I

180

1. Banach Spaces

1.3.3 T h e H a h n - B a n a c h Theorem

1.3.3.1

Theorem

( 0 ) (Hahn 1927, Banach 1929) Let F be a subspace of

the normed space E . Take y' E F ' . Then there is a continuous linear extention x' of y' to E with IIx'll- Ily'll.

Case 1

]K = IR

Define

p:E

>~,

x,

>lly'llllxll,

Then

p(x + y) = Ily'll IIx + yll ~ Ily'll(llxll + Ilyll) ~ p(x) + p(y), p(~x) = Ily'll II~xll- Ily'llallxll - ~p(x) for every x, y E E and a E IR+. Moreover

y'(x) ~ Ily'll Ilxll = p(x) for every x C F . (Proposition 1.2.1.4 a)). By the Banach Theorem, there is a continuous linear extension x' of y' to E with

x'(x) < p(~) for every x C E . Then

x'(x) ~ Ily'll IIxll and

-x'(x) : x ' ( - x ) ~ Ily'll II- xll = Ily'll IIxll and so

Ix'(x)l ~ Ily'll IIxll for every x e E . Hence x' is continuous and IIx'll _~ Ily'II. By Proposition 1.2.1.4 c), IIx'[I- Ily'll. Case 2

IK = ~

1.3 The Hahn-Banach Theorem

rey'

is a c o n t i n u o u s real l i n e a r f o r m on F .

a continuous

] R - l i n e a r f o r m z' on E

181

B y t h e a b o v e proof, t h e r e is

extending

re y' w i t h

IIz'[[ = [ire o Y'll-

Define

x' : E Then

>r

x p is l i n e a r ( L e m m a

x,

>z'(~)-iz'(iz).

1.3.1.5) a n d c o n t i n u o u s . B y C o r o l l a r y 1.2.1.6,

IIx'll = I1~ o x'll = I1~'11 = I1~ o y'll = Ily'll. Then re o x' (x) = z' (x) = re o y ' ( x ) ,

i m o x'(x) = - z ' ( i x ) = - r e

o y'(ix) = re o ( - i y ' ) ( x ' ) = i m o y ' ( x ) ,

x'(x) = re o x'(x) + i i m o x'(x) = re o y'(x) + i ira o y'(x) = y'(x)

I

for e v e r y x C F , i.e. x ~ is a n e x t e n t i o n of yl. Corollary

1.3.3.2

( 0 )

Let F be a finite-dimensional vector subspace of

the normed space E . Let G be a closed vector subspace of E with F N G = { 0 } . Let (x~)~I be an algebraic basis of F .

a)

There is a family (xt~)~, in E' such that x'~ vanishes on G for every c I and

x'~(~) - M for every L,A E I . b)

The map p "E

>E ,

x,

>

E 'x~(x)x~ L6I

is a projection of E onto F , vanishing on G. c)

F is a complemented subspace of E .

d)

If the map FxG

>E,

(x,y)~

>x+y

is surjective, then E is the direct sum of F and G and 1 - p projection of E onto G , vanishing on F .

is the

182

1: Banach Spaces

a) Put H:=F+G. Take ~ E I

and

x ~'F

)IK,

Ea)'x~'

>a,.

.XEI

By Lemma 1.3.1.1, there is a linear form y' on H such that y'IF = ~',

y ' l a = o.

K e r y ' is the vector subspace of H generated by C U {x~ I A E I\{~}}. It is thus a closed vector subspace of H (Corollary 1.2.4.3). Hence y' is continuous (Corollary 1.2.4.12 b =~ a). By the Hahn-Banach Theorem, there is a continuous linear extention x~' of y' to E . x ' vanishes on G and

z~(x~) = ~ !

for every A E I . b) p is linear and continuous. It vanishes on G and Imp c F . Moreover, given ~ E I ,

AEI

Hence I m p = F . Thus

tEI

tEI

~EI

for every x E E , so that pop=p. Hence p is a projection of E onto F . c) follows from b) and Murray's Theorem (Theorem 1.2.5.8). d) Since Kerp-

G,

E is the direct sum of F and G , and 1 - p vanishing on F (Murray's Theorem).

is the projection of E onto G, I

1.3 The Hahn-Banach Theorem

D e f i n i t i o n 1.3.3.3

( 0 )

183

Let E, F be normed spaces. Set

<., x'>y : E

>F ,

x,

> <x, x')y

~F

z,

~

for every (x', y) E E' • F and ', X~')y~

"E

~EI

(x, x~ LEI

for every finite family ((x',, y~))~e, in E ' • F . C o r o l l a r y 1.3.3.4 Then u belongs to s

( 0 ) Let E , F

be normed spaces and take u E / : ( E , F ) .

, F) iff there is a finite family ((x',, Y,)),E, in E' x F

such that

Assume u E s

F ) . By L e m m a 1.2.4.6, Ker u has finite codimension in

E . Thus by Corollary 1.2.5.6, there is a finite-dimensional vector subspace G of E such t h a t E - G | Ker u. Let (x~)~ei be an algebraic basis of G . By Corollary 1.3.3.2, there is a family ' (x~)~ei in E ' , such that for each ~ E I , x~' vanishes on K e r u and

x:(x~) = ~ for every 5, A E I , the m a p

p- E

~

E,

x,

~ ~<~,~'~/x~ eEI

is a projection of E onto G vanishing on Ker u , and 1 - p E onto K e r u vanishing on G . Given ~ E I , put y~ :-- ux~. Then

= upx u,1

p,x = u

for every z E E , i.e.

eEI

The reverse implication is trivial.

x:,x)=

is a projection of

184

1. Banach Spaces

Corollary 1.3.3.5

( 0 ) Let F be a vector subspace of the normed space

E . Take x E E with dF(X) > O. Then there is an x' E E' such that

IIx'll =

x' l F = o,

1,

x'(x) - dF(x).

Put a "= {c~x + Y l c~ E lK, y E F }

and let y' denote the linear form on G which vanishes on F and takes the value dF(x) at x (Lemma 1.3.3.1). Take (a,y) E IK • F with

I1~ + yll _~ 1,

a -r 0.

Then 1>

IIo~x+ yll-

1 --yll > Ic~ldF(x) = [c~dF(x)[ = ly'(~x + Y)I-

Io~1IIx

Hence y' is continuous (Proposition 1.2.1.1 d =v a) and Ily'll-< 1 (proposition 1.2.1.4 b)). Given y E F , dF(x) = y ' ( x -

y) ~

IlY'IIIIx- Yl

(Proposition 1.2.1.4 a)) and so dF(x) < ]lY'iidF(x),

1 < ]Iy'II,

Ily' = 1 -

By the Hahn-Banach Theorem, there is a continuous linear extention x' of y' to E with IIx'll = Ily'll-x' has the desired properties. I

Corollary 1.3.3.6

( 0 ) Let F be a vector subspace of the normed space

E . If F is not dense, then there is an x' E E'\{0}

which vanishes on F .

Let x E E \ F . Then dF(x) > O. By Corollary 1.3.3.5, there is an x' E E ' , vanishing on F , such that x'(x) = dF(x).

Then x' -~ 0.

Corollary 1.3.3.7 dual E' is.

I ( 0 )

The norrned space E is separable whenever its

1.3 The Hahn-Banach Theorem

185

Let (x'~)ne~ be a dense sequence in E ' . For each n E IN, there is an xn C E with

llx,~ll ___1,

1

l='(x.)l _> ~llx'nll

(Proposition 1.2.1.4 b)). Let F be the closed vector subspace of E generated by (Xn)neIN and let x' be a continuous linear form on E vanishing on F . (x~)~e~ contains a subsequence (x~,)nc~ which converges to x'. Then I

I

I

-2 Ilxk. II _< Ixko (~k~

I(xk~ xko)-(xk~ , x'>l- I(xk. , x'k~ x')l < II~k.' -*' II,

IIx'll ~ IIx'- x%~II-4-Ilxg= II ~

311x'k=

-

x'll

for every n E IN. Thus

IIx'll _< lim 31Ix'~o - m ' I I - 0 . Thus x ' = 0 and F = E (Corollary 1.3.3.6). Hence E is separable (Corollary

1.1.5.5).

1

Remark. The separability of E does not imply that of E' as is shown by taking E := ~1 (Examples 1.1.2.5, 1.2.2.3 d), and 1.1.2.2). C o r o l l a r y 1.3.3.8

( 0 ) Let E be a normed space and take x E E \ { 0 } .

a)

There is an x' e E' with IIx'[I = 1, x'(x) - lixll.

b)

Ilxl] =

sup Ix'(x)l = sup Ix'(x)l. XI E E l #

x IEE t

lix'll=l a) follows from Corollary 1.3.3.5. b) Given x' c E ' # ,

i~'(x)i < llx'll iizli < iixli (Proposition 1.2.1.4 a)) and so sup Ix'(x)l ( I l x l l . x, E E , #

By a), there is an y' C E' with

ily'll =1,

r

Hence Ilxll _< sup ]x'(x)] < x' E E'

IIx'ii=l

sup ]x'(x)] < ]]x]]. x, E E , #

i

186

1. Banach Spaces

C o r o l l a r y 1.3.3.9 y in E .

0

I

~

\

) Let E be a normed space and take distinct x and

Then there is an x t E E '

with

9'(~) # ~'(y) Since x - y ~ 0, the assertion follows from Corollary 1.3.3.8 a). C o r o l l a r y 1.3.3.10

(0 )

I

oo

Let E be a Banach space and ~ tnxn a power n--O

series in E . If there is an r > 0 smaller than the radius of convergence of the above power series such that E

olnxn -- 0

n--O

for every c~ E U~(O), then Xn --~ 0

for every n = IN u {O} .

Take n E I N U { 0 }

and x ' E E ' . T h e n

for every c~ E U~(0) (Proposition 1.2.1.16), so that

::,:'(~n) = o by a classical result of function theory. Since x' is arbitrary, Xn = 0

(Corollary 1.3.3.8 b) ). C o r o l l a r y 1.3.3.11

I Let E , F

be Banach spaces. Take u E s

Let U

be a domain of IK and take f : U--4 E .

a)

If f

is differentiable at (~ E U then u o f is differentiable at (~ and ( u o f ) ' ( o l ) = uf'(c~) .

b)

f

is constant iff f

is differentiable and f ' = O.

1.3 The Hahn-Banach Theorem

187

a) is trivial. b) The necessitiy is trivial. Take a, fl C V and assume that f(c 0 ~ f ( f l ) . By Corollary 1.3.3.9, there is an x' c F ~ with x ' ( f (ol)) :/: x ' ( f (,3)) .

By a), x' o f is differentiable, and by the above, its derivative does not vanish identically. Hence, by a), the derivative of f does not vanish identically.

I C o r o l l a r y 1.3.3.12

Let E be a normed space and p a norm on E' equivalent

to the canonical norm on E ' . Put

q(x) := sup{Ix'(x)l lx' e E',p(x') < 1}. Then

E

~ JR+, x',

~ q(x)

is a norm of E equivalent to the initial norm of E . Let Eq (resp. E p ) denote the vector space E

(resp. E ' ) endowed with the norm q (resp. p ). If p is

lower semicontinuous on E'E, then E~ is the dual of Eq.

Since p is equivalent to the canonical norm on E t , there is an a > 0 such that 1

- p ~ II-II ~ ~p. Hence

1

--q --< II" II _< t~q. ol

(Corollary 1.3.3.8 b)). In particular, q is finite and x E E , q(x) = 0 ~

x = O.

Take x, y E E . Then ]x'(x + y) = Ix'(x) + x'(y)] < Ix'(x)] + Ix'(y)l < q(x) + q(y)

for every x' E E ' , p(x') < 1, so t h a t

q(x + ~) < q(~) + q ( y )

188

1. Banach Spaces

Moreover,

q(j3x) = sup{]x'(~x)] ] x' E E ' , p(x') < 1} = = [/3[sup{[x'(x)[ ] x' E E', p(x') <_ 1} --[/3[q(x) for every /3 E IK. Hence q is a norm on E equivalent to the initial norm of E. Now assume that p is lower semicontinuous on E ~ . Put p"E'

)IR+,

;sup{Ix'(x)I I x E E ,

x':

q(x) < 1},

A' "= {x' E E' [ p(x') < 1}. Is is easy to see that p' <_ p. Take x' E E ' \ A ' . Since A' is a closed absolutely convex set of E~ (Proposition 1.2.7.2), there is an x E E with sup rey'(x) < rex'(x) y~ E A'

(Proposition 1.3.1.8). Hence

q(x) = sup ] y ' ( x ) l - sup rey'(x) < rex'(x) < p'(x')q(x), y~ E A'

yl E A'

p'(~') > a.

Take y' E E ' \ { 0 } and let c~ E ]0,p(y')[. Then !y, E E ' \ A ' so

by the above considerations. It follows successively that p'(y') > . .

p'(y') ___ p ( y ' ) ,

p - p,.

and E~ is the dual of Eq.

Remark. The final assertion does not hold if p is not lower semicontinuous (see Example 1.3.1.16 c)). (E. Helly) Let E be a normed space, (xP~)~e, a finite linearly independent family in E t, and ~ > O. Given a E ]K l , the following are equivalent:

C o r o l l a r y 1.3.3.13

1.3 The Hahn-Banach Theorem

a)

189

For every c > O, there is an x E E such that I1~11 ~ ~ + ~ and !

x,(x)-

a~

for every t E I . b)

] E atat] < a]] E atx'tl] tel

for every (at)tEl E IK I .

tel

a ~ b. Take ~ > 0 and x satisfying a). Then

, tel

= I tel

I

t6I

tel

I <-

tel

Since c is arbitrary,

tel

tel

b ~ a. Put

p:]K I---+IR+,

b,

>inf{flE]R+lbE/3A

}.

Since A is absolutely convex, p is a norm. Endow IK I with this norm. By Corollary 1.3.3.8 a), there is an x' E (IKI) ' such that x'(a)

II~'ll = 1,

= p(a)

.

Take (at)rE1 E IK I such that

tEI

for every b E IK'. By b) (and Proposition 1.2.1.4 b)),

p(a) - x'(a) = E a t a t

<_ all E a t x : l I

tel

tel

= sup l E bEA

- sup{] E a t x [ ( x ) l

]x 6 a E # } =

tel

atbt[ = llx'[l = l .

tel

Hence a E (1 + c)A for every c > O, and a) now follows.

I

190

1. Banach Spaces

P r o p o s i t i o n 1.3.3.14

Let E , F

be normed spaces.

II<',x'>Yll =

IIx'll IlYlI-

~)

(~',y) E E' x r ~

b)

If E ~ {0} then F is complete iff s

F) is complete.

a) follows from

II<-,x'>yll = ~up II<=,x'>yll = ~up I<=,x'>l Ilyll--IIx'll xEE#

xEE#

Ilyll.

b) Assume f_.(E,F) complete and let (Yn)ne~ be a Cauchy sequence in F . Take x E E \ { 0 } . By Corollary 1.3.3.5, there is an x' E E' such that

IIx'll

= 1,

<x,x'> = 1.

By a), ((',x')yn)ne~ is a Cauchy sequence in L ( E , F ) . Denote by u its limit. Then

ux = lim <x, x')yn = lim yn. n ----~oo

Hence (Yn)ne~ converges and F is complete. The converse implication follows from Theorem 1.2.1.9 b).

I

1.3 The Hahn-Banach Theorem

191

1.3.4 T h e T r a n s p o s e of an O p e r a t o r

Definition 1.3.4.1 take u C s

( 0 )

(Banach 1 9 2 9 ) L e t E , F

be normed spaces and

F ) . The map u " F'

> E',

y' J

> y' o u

is called the transpose (operator) o f u .

T h e o r e m 1.3.4.2 a)

( 0 ) Let E, F be normed spaces and take u C s

F).

u' is the unique map F' --+ E ' , such that

for every (z, y') e E x F'

b)

u' e s

E'),

I1~'11 =

Ilull.

c) u is determined by u'. a) Given (x,y') c E x F ' , (~, ~'y') = (x, y' o ~) = y' o ~(x) = y'(~x) = ( ~ , y'). Take f " F' --+ E ' , such that (~, f(y')) - ( ~ , y') for every (x, y') E E x F ' . Then (x, f(y')) = (x, ~'~') for every (x, y') e E x F ' , s o that f(y') = u'y' for all y' e F'. Thus f = u'. b) Take x', y' C F' and c ~ , / ~ C I K . B y a ) (x, ~'(~x' +/~y')) - ( ~ , ~x' +/~y') - ~(~x, ~') + Z ( ~ , y') -

= . ( z , ~'~') +/~(~, ~'y') - (~, .~'x' + Z~'~') for every x C E . In other words,

Hence u' is linear.

192

1. Banach Spaces

Given y' E F ' ,

II~'y'll = Iiy'o ~11 < Ily'll ii~ll (Corollary 1.2.1.5), and so u' is continuous (Proposition 1.2.1.1 d ==> a) and

il~'il < il~il. Take x E E with ux ~ O. By Corollary 1.3.3.8 a), there is a y' C F' such that

ily'll = 1,

y'(~x) - ii~ll 9

y'o ~(~)-

(x, ~'y') < i~ll il~'y'li < il~ll ii~'l[

Thus

li~xll = y'(~)-

(Proposition 1.2.1.4 b) ), so

II-II <_ ii-'Ii,

I-li = II-'ll. !

c) follows from a) and Corollary 1.3.3.9.

Corollary 1.3.4.3

( 0 ) If ~, r a~ no,~ed ~pac~, th~n th~ map s

u,

~ f_,(F',E'),

> u'

(Theorem 1.3.4.2 b) ) is linear and continuous.

Take u, v c s

F ) and c~, fl C IK. Then

= ~(x, ~ ' y ' / + Z(~, ~'y') - (~, ~ ' y ' + Z~'y') = (~, ( ~ ' + Zv')y') for every (x, y') e E x F' (Theorem 1.3.4.2 a)). Thus

~u' + fl,'=

(~ + 9,)'

(Theorem 1.3.4.2 a) ). Hence the map s

>s

E') ,

u ~ ~ u'

is linear. By Theorem 1.3.4.2 b), it is continuous. C o r o l l a r y 1.3.4.4

( 0 ) If E is a normed space, then (1E)' -- 1E,.

m

1.3 The Hahn-Banach Theorem

C o r o l l a r y 1.3.4.5

193

( 0 ) Let E , F , C be normed spaces and take u e ~(E, F),

~ 9 L(P, a ) .

Then V 0 U) I ~

~t t 0 V ! .

Given (x, z') C E x G',

(vo~(~),z') = (wx, z')= (~,v'z')= (~,~'~'z')= (~,~' o~'(z')) (Theorem 1.3.4.2 a)). Thus u' o ~ ' =

(~ o ~)'

(Theorem 1.3.4.2 a) ). C o r o l l a r y 1.3.4.6

m The transpose of a projection is a projection.

This is an immediate consequence of Corollary 1.3.4.5. C o r o l l a r y 1.3.4.7

( 0 ) Let E , F

be normed spaces and take u E s

If u is an isomorphism (isometry), then u' is an isomorphism (isometry).

Let u be an isomorphism and let v := u -1 . Then uov=

1F,

you=

1E.

Thus v ~ou I = 1F,,

u ~ o v ~ - 1E,

(Corollary 1.3.4.5, Corollary 1.3.4.4), i.e. u' is an isomorphism. If u is an isometry, then for y' E F ' , ll~'y'll = sup I(~,~'y')l = s ~ p I ( ~ , y ' ) l xEE#

xcE#

= sup I(y,y')l = Ily'll yEE#

(Proposition 1.2.1.4 b), Theorem 1.3.4.2 a)), i.e. u' is an isometry. Remark.

The reverse implication, u' isomorphism (isometry) ==~ u isomorphism (isometry),

holds whenever E and F are complete (Corollary 1.4.2.5).

R

193

1. Banach Spaces

C o r o l l a r y 1.3.4.8

(2)

(3)

Let E , F , G , H

be normed spaces. Take

v E / : ( G , H ) and let ((y~,zt))te, be a finite family in F ' •

u E s

G.

Then

(E (, ~:>z~) o ~: E ( , ~,~:>z~, v o ( E (, ~:>z,) - E (, ~:>~z~ tEI

Given x E E

tel

tel

tel

and y E F ,

tel

tEI

tcl

",

-

Yt)

t)x

eEI

(Theorem 1.3.4.2 a) ), and

v( ~ < , ~:>z~)~ =

~( E<~, ~:>z,) = ~<~,~:>vz,

tel

tel

: ( E ~ , ~:>vz,)~

tel

tel

which proves the assertion, C o r o l l a r y 1.3.4.9

i

( 0 ) Let E and F be normed spaces. Take

9 L(F,E'). Then the following are equivalent:

a)

The map u" F~ --~ ErE is continuous.

b)

There is a v E / : ( E , F )

such that u - v ' .

The operator u of b) is unique and is called pretranspose o f u.

a =v b. Take x E E . By a), the linear map F~ ---+ IK,

y',

~ (x, uy')

is continuous. By Corollary 1.2.6.5, there is a vx E F such that

(x, ~y') - (~x, y') for every y' E F ' . It is obvious that v" E --+ F is linear. If x E E # , then

I(~, y'>l = I(x, ~y')l ___ Iluy'll _< I~11 Ily'll for every y' E F ' . By Corollary 1.3.3.8 b),

1.3 The Hahn-Banach Theorem

195

Hence v is continuous. By Theorem 1.3.4.2 a), v' = u. b ~ a. By Theorem 1.3.4.2 a),

(~, ~y') = (vx, y') for every (x, y') C E • F ' . a) now follows. The uniqueness of v follows from Theorem 1.3.4.2 c). Theorem

1.3.4.10

(Banach-Stone) Let S , T

be compact spaces. Given

u E f_,(C(S),C(T)), the following are equivalent:

a)

u is an isometry.

b)

There is a h o m e o m o r p h i s m f : T - + S and a y C C(T) such that ly(t)l = 1

for every t C T and

~ = y(x o / ) for every x C C ( S ) .

a => b. We identify C ( S ) ' , C(T)' with the Banach space of Radon measures on S and T , respectively. By Corollary 1.3.4.7, u' is an isometry. Given s c S (resp. t e T ) , let as (resp. at) denote the Dirac measure on S (resp. on T ) at s (resp. t). Take t c T . Then at is an extreme point of C(T) '# (Example 1.2.7.14) and so u'at is an extreme point of C(S) '# . By Example 1.2.7.14, there are f (t) e S and y(t) E IK such that

ly(t)l = 1,

u'at = y(t)a/(t) .

f is injective, for otherwise u' would not be injective, f is also surjective, since u' is surjective. Take x C C(S). Then y ( t ) x ( f ( t ) ) = (x, y(t)5/(t)) = (x, u'ht) = (ux, at) = u x ( t ) .

We see from this relation that y and f are continuous. Since f is bijective, it is a homeomorphism. Moreover, ~

b =~ a is easy to see.

= v(~ o f).

I

196

1. Banach Spaces

Example

Take p 9 [1, c~[, and let q be the conjugate exponent of

1.3.4.11

p. Let u be the right (left) shift in ~ . Then u' is the left (right) shift in gq, where tq is identified with (tP)' (Example 1.2.2.3 d) ). Let v be the left (resp. right) shift in t~q . Then (x)

(x)

y> =

= n=2

=

oo

(resp. (ux, y> - E

o(3

xn+,Yn = E

n=l

for every (x, y)

9 e~

vy>

n=l

xnYn-1 -- (X, vy))

n=2

• t~q and the assertion follows from Theorem 1.3.4.2 a). m

Let S, T be sets. Take p, q 9 [1, oo] t3 {0} and let p' and q' be the conjugate exponents of p and q, respectively. Take

E x a m p l e 1.3.4.12

k 6 gP'q'(S,T) and put N

n

k" gq(T)

~ gP(S),

x,

~ kx

(Proposition 1.2.3.~ b)). If either S or q is finite, then ~ ) x = kx for every x' 6 gP'(S), where s (Example 1.2.2.3 b) ).

has been identified with a subset of gP(S)'

Given x c eq(T),

tET

t6T

sES

tET

s6S

(Theorem 1.3.4.2 a) ).

m A

Remark.

If S and T are finite, then k is the matrix associated to k and the N

transpose of k is the matrix associated to (k)'.

1.3 The Hahn-Banach Theorem

197

E x a m p l e 1.3.4.13 Take n C IN, and let u 9IK n --+ IK 2 be a linear map, with associated matrix [aij]ie~2,je~,. If we endow IK ~ and IK 2 with the Euclidean norms, then

]alj +

]ltt]2--~ j=l

]a2j

+

j=l

n

n

,~2

Elalj]2+Ela2j]2~ j=l j=l / -4 j=l

lalJ]2} ( E

la2J12} - I E

/

/

\j=l

alj-~2jl2 j=l

= 1(~-~2 ( IOLljl2--]-]0/2j]2) -'~j--1

)2 j=l

) i,j=l

By Example 1.3.4.12, the matrix associated to u" IK'2 -+ I[4n is the transpose of the matrix [O~ij]ielNe,jelNn, and by Theorem 1.3.4.2 b), I~'11 = I1~11 m

The assertion follows now from Example 1.2.2.7 d).

E x a m p l e 1.3.4.14 ( 1 ) Let S , T be sets, p,p' weakly conjugate exponents, and q, q' be conjugate exponents. Take k C t~P"q(s, T) and put U

[J

(Proposition 1.2.3.8 b) ). If T or p is finite, then CI !

(;)'~' = kx f o r e v e r y x t C ~q' ( T ) , where ~q' ( T )

(Example 1.2.2.3 b) ).

has been identified with a s u b s e t of gq(T)'

198

1. Banach Spaces

Given x E tP(S),

sES

= Ex(s)( sES

tET

E

sES

k(s, t)x'(t)) = (x, kx'}

tET

(Theorem 1.3.4.2 a) ).

I

Example 1.3.4.15

Let S , T be locally compact spaces, f 9S ~ T a proper continuous map and put u ' g o ( T ) ---+go(S),

x,

~xo f .

Then for each p E .Mb(S), u'# is the image f ( # ) of p (Example 1.2.2.10).

For the proof see, for example, N. Bourbaki, Integration (1956), Ch. V, w I

1.3 The Hahn-Banach Theorem

199

1.3.5 P o l a r Sets D e f i n i t i o n 1.3.5.1

( 0 ) Let E be a normed space, F a vector subspace

of E , and G a vector subspace of E ' . Put

(polar of F),

F ~ := {x' e E' I x'lF = 0} ~

(prepolar of G).

:= ['~ Kerx' x'cG

F ~ is a closed vector subspace of E~ and ~ of E . We have

{0}~

',

E~

is a closed vector subspace

~ ~

(the last equality follows e.g. from Corollary 1.3.3.8 a) ). P r o p o s i t i o n 1.3.5.2

( 0 )

Let E be a normed space, F a closed vector

subspace of E , and q : E -+ E / F

the quotient map. Then

Im

q' = F ~

and the map (E/F)'

>F ~

y',

> q'y'

is an isometry. In particular, F ~ is a dual space.

The assertion follows from Proposition 1.2.4.7. Remark.

II

( E / F ) ' and F ~ are frequently identified using the above isometry.

P r o p o s i t i o n 1.3.5.3

( 0 ) Let E be a normed space, F a vector subspace of

E ' , and (x~)~ei a finite family in E such that no nontrivial linear combination of (x~)~i belongs to ~

Then there is a family (x'~)~i in F , with

x:(~) = ~ for every ~, A C I .

200

1. Banach Spaces

We prove the assertion by induction on the cardinality of I . Take A E I and put J "- I\{A}. By the inductive hypothesis, there is a family (Y~)~cz with

for every t,# C J . Given t C I , put

~-F

~IK,

x',

>x'(x~).

Assume that N Ker ~ C Ker ~ . tCJ

(If J - 0, replace the intersection by F . ) Then, by Lemma 1.2.6.3, there is a family (c~)~ej in IK with XA -- E

~

"

tEJ

Thus

tea

tCJ

for every x' C F and so E

OF.

LCJ

This contradicts the hypothesis of the proposition. Hence, we can find an x'E (NKer~)\Ker~. Now put 1

x~! .. _-_ ~x,(x~) x ! and !

.

f

I

!

for t E J . The family (x'L)~eI has the required properties.

I

1.3 The Hahn-Banach Theorem

P r o p o s i t i o n 1.3.5.4

( 0 ) Let E be a normed space and F ~ is the closure of F in ErE .

201

a vector

subspace of E ' . Then (~

Let F be the closure of F in E E . It follows from F C (~176 that F C (~176 Take x' C E ' \ F . We prove that x' r (~176 Assume the contrary. Let A be a finite subset of E with

CI {y' < E l

-x'(x)I < I}

xEA

(Proposition 1.2.6.2). Let G be the (finite-dimensional) vector subspace of E generated by A. Let (xL)~e~ be an algebraic basis of G such that {x~ [ C I , x~ E ~ is an algebraic base of G A ~ Put S := {~ C I Ix~ q~ ~ By Proposition 1.3.5.3, there is a family (x[)Lej in F with 9: ( ~ ) = 5 ~

for ~, A c J . Put

cEJ

Then y'(~) = x'(~)

for every ~ C J and y'(z~) = 0 = ~ ' ( ~ )

for every ~ E I \ J .

Hence y ' - x ' contradiction we sought. C o r o l l a r y 1.3.5.5

( 0 )

Let E be a normed space, F a subspace of E'

closed in ErE, and q " E ~ E / ~ (E/~

= 0 on G, so that y' r F , which is the I

the quotient map. Then the map '--+ F,

x'~

~ q'x'

is an isometry.

The assertion follows immediately from Proposition 1.3.5.2 and Proposition 1.3.5.4. I

202

1. Banach Spaces

1.3.5.6

Corollary

( 0 )

Let E be a normed space. Then a subspace of E'

is a dual space whenever it is closed in E~E .

The assertion follows immediately from Corollary 1.3.5.5. Proposition

1.3.5.7

I

( 0 ) Let F be a vector subspace of the normed space

E . Then

~176 = F. The inclusion FC~

~

is trivial, so F C ~176 Assume that F :/: ~176 Then there is a y' E E' with

y'l F - O ,

y' I ~176 ~ 0

(Corollary 1.3.3.5). But then y' E (F) ~ = which is a contradiction. Hence r =

Proposition 1.3.5.8 s

( 0 )

F ~

= (~176176 (Proposition 1.3.5.4),

o(po),

Let E , F

m be normed spaces and take u E

F) . Then

Keru'=(Imu) ~

Keru=~

Imu=~

Take (x, y') E E x F ' . Then (~x, y') - (~, ~'y')

(Theorem 1.3.4.2 a) ). It follows immediately from this, that y'EKeru'~y'E

(Imu) ~

1.3 The Hahn-Banach Theorem

203

so t h a t Ker u' = (Im u) ~ and Im u = o((im u) ~ = ~

u')

(Proposition 1.3.5.7). By the above equality and Corollary 1.3.3.8 a), it further follows t h a t x E Ker u r

x E ~(Im u').

Hence Ker u = o (Im u'). Corollary

1.3.5.9

( 0 ) Let E , F

I

be normed spaces and take u E f _ . ( E , F ) .

Then u' is injective iff Im u is dense.

u' is injective

r

K e r u ' = {0} r

~

= F ~

Im u = F

(Corollary 1.3.3.8 a), Proposition 1.3.5.8). Remark.

I

The injectivity of u does not imply t h a t u' i~ surjective, as the

inclusion m a p t~1 --+ co shows. Corollary

1.3.5.10

( 0 )

Let E, F be normed spaces. Take u E f_.(E, F)

such that I m u is closed. Let q: F ~ F / I m u be the quotient map. Then

Im q~ = Ker u ~ and the map

(F/Imu)'

> Keru',

x'. ~ ~ q'x'

is an isometry.

We have Im q' = (Im u) ~ = Ker u' (Proposition 1.3.5.2, Proposition 1.3.5.8), and the assertion follows from Proposition 1.3.5.2.

I

204

1. Banach Spaces

C o r o l l a r y 1.3.5.11

Let E be a normed space and p a projection in E . Then

p~ is a projection in E ~ with

Kerp' = (Imp)~

Imp' = (Kerp)~

By Corollary 1.3.4.6, p' is a projection, and by Proposition 1.3.5.8, Kerp' = (Imp) ~

Imp' C (~

~

(Kerp) ~

Take x' E (Kerp) ~ . Then C Kerp

x-px

so that

(x, p'x') = (px, x') = (x, ~') for every x C E (Theorem 1.3.4.2 a) ) and (Kerp)~

x'=p'x'EImp',

Theorel~ 1.3.5.12

I m p ' = (Kerp)~

( 0 ) L~t r b~ a ~ p a ~

m

4 th~ ~o~m~d ~pac~ E a~d

u : F --+ E the inclusion map.

a)

Imu'=F'.

b)

Keru' = F ~

c)

The factorization E ' / F ~ ~ F' of u' through E ' / F ~ is an isometry.

d)

I f G is a closed vector subspace of ErE and v : ~ map, then the f a c t o r i z a t i o n E ' / G isometry. In particular, E ' / G

--+ (~

-+ E is the inclusion

of v' through E ' / G

is a dual space.

a) follows from the Hahn-Banach Theorem. b) follows from Proposition 1.3.5.8. c) Let v be the factorization of u' through E ' / F ~ . Then

Ilvll = If~'ll = II~lf _< 1 (b), Proposition 1.2.4.7, Theorem 1.3.4.2 b)). Take X ' E E ' / F ~ . Then

IlvXll ~ Ilvll IIX'll ~ IIX'll (Proposition 1.2.1.4 a)). Take x' E X ' . Since

is an

1.3 The Hahn-Banach Theorem

205

x ' l F - x' o u = u'x' = v X ' ,

it follows that

<x,x'> = <x,vX'> ~ IlvX'll Ilxll for every x E F (Proposition 1.2.1.4 a) ). Hence

IIx' FII ~ IlvX'll. By the H a h n - B a n a c h theorem, there is a y~ C E ~ with

y'IF- x' F,

IlY'II- Ilx'lFII.

Then y ~ - x ~ C F ~ i.e. y~ C X ~. Hence IIX'll ~-IlY'II = IIx'IFII-~ IIvX'll,

IlvX'l = IIX'II 9

By a), v is surjective. Hence v is an isometry. I

d) follows from c) and Proposition 1.3.5.4. Proposition

1.3.5.13

( 0 )

Let T

be a compact space and let C(T)'

identified with the Banach space of Radon measures on T . Let ~ subspace of C ( T )

be

be a vector

such that x-2, x y c jc f o r every x , y E .T and let It be an

extreme point of jco A C(T) '#

Then the f u n c t i o n s of ~

are constant on the

support of It.

We may assume that # ~ 0. Let x be a positive real function in ~ . Assume x is not constant on Supp #. Put

1

y:=-x,

ct

u := y . # .

Then

~#o,

~-~#0,

and I L'II + I # - u l ' - / y d l # +

f(l-y)dlp= /d,#,-,,#ll-I

(Proposition 1.2.7.11). Then zy, z(1 - y) e 9r , so t h a t

206

1. Banach Spaces

(z, u) -- f

zy d# = O ,

(z,p-u>=fz(1-y)dp=O for every z E .T'. Hence u,#1

1

I1~11 v, I1~- vii

u E .T~

( I t - u) e .T ~ fq C(T) '#

Since 1

1

and since # is an extreme point of .To N C(T) '# , 1 I~11 ~ = ~ '

y -I1~11 on S u p p # ,

9 = ~11~11 on S u p p l .

Hence x is constant on S u p p # . Now let x be an arbitrary function in .T and take s, t E Supp # with

x(~) # ~(t). Since x~ is a positive real function in .T, it follows from the above that x~ takes the same values at s and t. P u t / z~

if x ( s ) z ( t ) = 0

Y

I x ~ - ~(~)~1 '

if ~(~)x(t) # 0.

Then y is a positive real function in .T taking different values at s and t, which contradicts the above result. Hence x is constant on S u p p # . Theorem

1.3.5.14

I

( 0 ) Let T be a compact space and .T a vector subspace

of C(T) such that x-2, xy E .T for every x, y E .T. Put

S "-- N xl(O) xE.~" and /

t" ~ for s, t G T \ S .

\

( x E .T ~

xIs ) -- x I t ) )

Let ~ denote the set of x E C(T) which vanish on S and for

which z(s) = z(t) whenever s, t E T \ S

satisfy s ~ t. Then .T = G.

1.3 The Hahn-Banach Theorem

207

Take x E g . Identify C(T)' with the Banach space of Radon measures on T . Let # be an extreme point of ~-~

C(T) ~#. By Proposition 1.3.5.13, the

functions in ~" are constant on S u p p # . Hence there is a y E $" with x = y on S u p p # and so (~, , ) = ( y , , )

= 0

9v~ is a closed vector subspace of C(T)~c(T). Hence by the Alaoglu-Bourbaki Theorem, yo N C(T) '# is a compact set of C(T)~c(T). By the K r e i n - M i l m a n Theorem~

c.. tT~,# ~- j "

'

o n t ~

the extreme points of ~'~ N C(T) '# . By the above, (x,#) = 0 for every # C 9v~ A C(T) '# . Hence

~ e~176 =7 (Proposition 1.3.5.7) and

gc.F.

I

The reverse inclusion is easy to see.

Remark.

The idea of using extreme points for such denseness problems is due

to de Branges (1959). C o r o l l a r y 1.3.5.15

Let T be a locally compact space and 3z be a

( 0 )

vector subspace of Co(T) such that: 1)

If x, y c ~ , then xS, xy c J~ ;

2)

Given distinct s, t e T there are x, y e .~ such that x(s)y(t) ~ x ( t ) y ( s ) .

Then :7z is a dense set of Co(T). Let T* be the Alexandroff compactification of T and extend each function in Co(T) by setting it equal to 0 at the Alexandroff point of T . By 2), S of Theorem 1.3.5.14 contains only the Alexandroff point of T , and the equivalence classes of ~ are one point sets. Corollary

1.3.5.16 ( 0 )

be a compact space and ~

(Weierstrass-Stone Theorem,

I

1885, 1937).

a vector subspace of C(T) such that:

Let T

208

1. Banach Spaces

1) x , y E iF => x h , xy c iF. 2)

Given distinct s, t e T , there are x, y e iF with x(s)y(t) =/=x ( t ) y ( s ) .

Then iF is a dense set of C ( T ) .

C o r o l l a r y 1.3.5.17

I

( 0 ) Let T be a set and iF a closed vector subspace of

g ~ ( T ) such that x-2, xy E iF for every x, y E jc. Take x C iF and f E C(x(T)) such that o e x ( T ) , ~ r J= ~

S(O) = o.

Then f o x C iF.

Let G be the smallest vector subspace of C ( x(T)) such that: 1) the function x(T)

> IK,

a,

>

is in G, 2) e~--(-~C G whenever eT C iF, 3) g-~, gh G ~ for every g, h C G. Then f ox C iF for every f C G .By Theorem 1.3.5.14, f belongs to the closure of ~ in C ( x ( T ) ) . Hence there is a sequence (f~)ne~ in ~ converging uniformly to f . Then (fn o X ) ~

converges uniformly to f o x. Since iF is closed and

fn o X C iF for every n C IN, it follows that f o x E iF.

C o r o l l a r y 1.3.5.18

1

Let T be a compact space and iF a vector subspace of

C(T) such that xy C iF for every (x, y) C C(T) x iF. Put s ::

Nx

(0) 9

xC.~

Then

7-

{~ e c(T) I ~lS - 0 }

Given distinct s, t c T \ S , there is an x c iF with

9(~) ~ ~(t). The assertion thus follows from Theorem 1.3.5.14.

I

1.3 The Hahn-Banach Theorem

Proposition

1.3.5.19

Let T

be a locally compact space and ~

209

a vector

subspace of C(T) such that: 1)

5 x , xy C .~ for every x , y E .T.

2)

For distinct s, t E T , there are x, y C .~ with

9(~)y(t) - ~(t)y(~) # 0 3)

e T C . ) E" .

Let 9 be the coarsest topology on C(T) for which the functions C(T)

> IK,

~, f x d #

x~

are continuous for every bounded Radon measure # on T . Then jz is dense in C (T) with respect to ~ . Let /~T be the Stone-(~ech compactification of T . We consider the functions in .~ to be extended continuously on f i t and put ~

/~T

~ IK 7 ,

t,

>, (x(t))xej:

Then ~ is continuous and, by 2), it is injective on T . Let x c C(T) have compact support. Let (#~)~e~ be a finite family in .s

and take c > 0. T has a compact set K such that Supp x c K

and I#~I(T\K) <

2(1 + Ilxll)

for every ~ C I . Then the m a p

r

t,

>~(t)

is a homeomorphism. Hence x l K - y o r for some y C C ( ~ ( K ) ) . By Tietze's Theorem, y can be extended to a continuous function on ~(/~T) such that

Ilyll- I1~11 Then

210

1. Banach Spaces

ly o ~ldl~l

=

<__ Ilyll I P ~ I ( T \ K ) _< IIxll2( 1 + Ilxll) <

T\K for every e E I . Take s, t E / 3 T such t h a t z(s) = z(t) for every z C $ ' . Then ~(s) = ~(t) so t h a t o ~(~) = ~ o ~(t).

By 3) and Theorem 1.3.5.14, there is a z C 9v such that

Ily o ~ - zll <

2 (1 + }--~ IIpLII) " eEI

Then

E

C

2 I+~II#LII LEI

for every ~ E I . Hence x is in the closure of $" with respect to ~ . Since the set of functions of C(T) with compact support is dense in C,(T) with respect to ~', it follows t h a t .~ is dense in C(T) with respect to ~s

1

1.3 The Hahn-Banach Theorem

211

1.3.6 T h e Bidual Definition 1.3.6.1

( 0 ) Let E be a normed space. The dual of E' is called the bidual o f E and is denoted by E" (Hahn, 1927). The dual of E" is called

the tridual of E and is denoted by E"'.

E x a m p l e 1.3.6.2 tric to g ~

( 3 ) If T is a set, then the bidual of co(T) is isome-

If we endow T with the discrete topology, then the tridual of

co(T) is isometric to the Banach space of Radon measures on the Stone-Cech c o m p ~ n

af T .

The first assertion follows from Example 1.2.2.3 d), e). The second assertion follows from the first one and Example 1.2.2.11. 1 T h e o r e m 1.3.6.3

( 0 ) (Hahn, 1927) Let E be a normed space.

a) <x,.) e E" and II(x,-)ll=llx][ for every x 9 E . b)

The map

E

~E",

z,

~(z,.)

is injective, linear, and continuous with norm 1 if E # {0}. It is called the evaluation of E and is denoted by j E .

a) (x,.} is linear and

Ilxll-- sup

I(x,x')l = II<x,.)ll

xlEE#

(Corollary 1.3.3.8 b), Proposition 1.2.1.4 b)). b) The map is obviously linear. The other assertions follow from a) and Proposition 1.2.1.1 d => a. m Corollary 1.3.6.4

( 0 ) A normed space is finite-dimensional iff its dual

is finite-dimensional.

The necessity was proved in Corollary 1.2.4.10. If the dual of a normed space E is finite-dimensional, then its bidual is finite-dimensional too. Thus, by Theorem 1.3.6.3 b), E is finite-dimensional, m Corollary 1.3.6.5

( 0 ) If E is a normed space, then ImjE is a dense set

of E~, and the map

E ----+ Im j E ,

x J >jEX

is an isometry. E is complete iff ImjE is closed in E " .

212

1. Banach Spaces

We have ~

= E~

{0},

so that (~

jE) )o = E" .

By Proposition 1.3.5.4, ImjE is dense in E~,. By Theorem 1.3.6.3, the map E

>ImjE,

z,~

~jEX

is an isometry. Hence E is complete iff Im jE is complete, and this is equivalent to I m j E being closed in E" (Corollary 1.2.1.10). 1 Remark.

a) E is frequently identified with ImjE via the above isometry.

b) The first assertion of the corollary will be strengthened in Corollary 1.3.6.8. C o r o l l a r y 1.3.6.6

( 3 ) If E , F (.,x[)yt

are normed spaces, then

)' =

(',jEYL)X'~

tel

for every finite family (( X't, Yt))tEl in

tel

g'

x F

.

We have

tel

tel

= E (x, x:)(Yt, Y') - E (x, x:)(y', jfYt) = tel

tel

tel

tel

for every (x, y') E E x F' and the assertion now follows from Theorem 1.3.4.2

a).

m

1.3 The Hahn-Banach Theorem

P r o p o s i t i o n 1.3.6.7

Let E be a normed space. Take x" E E " . Let

213

(Xtt)LeI

be a finite family in E' and take c > O. Then there is an x C E such that

I[xl[ < lix"[I + and



(x"

'

for every ~ C I .

Given (c~)~ci c IKr ,

~CI

tel

LE

and the assertion follws from Corollary 1.3.3.13 b ==~ a. C o r o l l a r y 1.3.6.8 is dense in ~ " #g ! Remark.

I

(Goldstine, 1938) If E is a normed space, then j E ( E #) I

9

The above result also follows from Proposition 1.3.1.8.

D e f i n i t i o n 1.3.6.9

( 0 ) Let E be a normed space, A a subset of E , and

A' a subset of E ' . We identify A with jE(A) (Theorem 1.3.6.3) and set AA, := (jE(A))A, (Definition 1.2.6.1). AA, i8 the set A endowed with the topology of pointwise convergence in A ' . If A' = E ' , then the corresponding topology is called the weak topology. The words weak and weakly used in conjunction with a topological term will signify that this term is considered with respect to the weak topology.

By Theorem 1.3.6.3, the weak topology is coarser than the norm topology. By Corollary 1.3.3.5, the weak topology is completely regular. P r o p o s i t i o n 1.3.6.10

Let A be a convex set of the normed space E . Then

A is closed iff it is weakly closed.

Let A be closed. We may assume that A ~ 0. Take x C E \ A . By Corollary 1.3.1.7, there is an x' C E' such that sup rex'(y) < rex'(x). yEA

Hence x does not belong to the weak closure of A and so A is weakly closed. The converse implication is trivial.

I

214

1. Banach Spaces

T h e o r e m 1.3.6.11

(I. Schur, 1920) Let T be a set. Every weak Cauchy se-

quence in gl(T) is norm convergent.

Let (x,)ne~ be a weak Cauchy sequence in gl(T), i.e. lim (xn, x') n---~oo

exists for every x' C gl(T)'. By Example 1.2.2.3 d), lim ~

xn(t)y(t)

tET

exists for every y E g~C(T) and the assertion now follows from Corollary 1.2.3.13.

I

P r o p o s i t i o n 1.3.6.12 ( 0 ) I.f E is a normed space, then the continuous and the weakly continuous linear forms on E coincide. I P r o p o s i t i o n 1.3.6.13 Let E be a normed space and E t . Then ~ is a weakly closed vector subspace of E .

G

a

vector subspace of

Take x' c E ' . Then x ~ is weakly continuous and so Ker x' is weakly closed. It follows from ~

M Kerx' x~EG

that ~

is weakly closed.

P r o p o s i t i o n 1.3.6.14

I ( 0 )

Let E be a normed space and F a vector

subspace of E . Then F~= ~ and F ~176is the closure of j E ( F ) in E~,.

Given (x, x') r E x E ' , (x, x') - (jEx, x')

and so F~- ~

Hence

and F ~176 is the closure of j E ( F ) in E~, (Proposition 1.3.5.4).

I

1.3 The Hahn-Banach Theorem

[

D e f i n i t i o n 1.3.6.15

(

0

)

Let E , F

215

be normed spaces and take u E

/:(E, F ) . The transpose of u' is called the bitranspose o f u and is denoted by U It .

Proposition

s

1.3.6.16

( 0 )

Let E , F

be normedspaces, and take u E

F) . Then u" o j E = j F o u .

For z E E , = <jEX, u'y'> = <X, u'y'> -- = <jF(UX), y'>

whenever y' E F' (Theorem 1.3.4.2 a)). Thus u" o jE(X) = u"(jEX) = jF(UX) = jR o u(x)

and !!

u ojE=jFou.

I

P r o p o s i t i o n 1.3.6.17 Let E be a normed space, F a subspace of E , and j " F -+ E the inclusion map. Then Im j" = F ~176 and the map F"

~ F ~176 y" l

>j " y "

is an isometry.

Let q 9 E' -+ E ' / F ~ be the quotient map and u 9 E ' / F ~ --+ F' the factorization of j' through E ' / F ~ (Theorem 1.3.5.12 a)). Then j' = u o q ,

so that j,, _ ql o u'

(Corollary 1.3.4.5). By Theorem 1.3.5.12 c) and Corollary 1.3.4.7, u' is an isometry. By Proposition 1.3.5.2, Im q' =

F ~176

216

1. Banach Spaces

and the m a p > F ~176 x" ,

(E'/F~ '

> q'x"

is an isometry. Hence Im j " = F ~176 , and the m a p F"

> F ~176 y" l

> j"y"

is an isometry, Proposition

i

Let E be a normed space, F a closed subspace of E ,

1.3.6.18

and q : E ~ E / F

the quotient map. Then Ker q" = F ~176

and the factorization of q" through E " / F ~176is an isometry. By P r o p o s i t i o n 1.3.5.2, I m q' = F ~

and the m a p

u : (E/F)'

>F ~

y',

> q'y'

is an isometry. Let j : F ~ ~ E ' be the inclusion map. Then

q'=jou so t h a t ql, ~ U' O

jl

(Corollary 1.3.4.5). Now Ker j ' = F ~176 and the factorization v of j' t h r o u g h E " / F ~176 is an isometry (Theorem 1.3.5.12 b), c)). Since u' is an isometry (Corollary 1.3.4.7), Ker q" = Ker j' = F ~176 . The factorization of q" t h r o u g h E " / F ~176is u' o v and therefore an isometry.

m

1.3 The Hahn-Banach Theorem

Proposition

1.3.6.19

(0)

217

Let E be a n o r m e d space and let .!

u := jE' o 3E" a)

fE o jE, = 1E, .

b)

u is the projection of E'"

onto I m j E ,

and

Ilull

~ 1; u is called the

c a n o n i c a l p r o j e c t i o n o f E'" (better: o f the t r i d u a l o f E ).

C) K e r u = (ImjE) ~ d)

E'" = (Im jE') @ (Im jE) ~

e)

If E is complete and q : E"

>E " / I m j E ,

r :Era

~ E'/ImjE,

are the quotient maps (Corollary 1.3.6.5), then r o q' is an isometry.

a) Given (x,x') c E x E ' , (x, j ~

o

jE,(X')) = (jEx, jE, X'} = {jEX, X') -- {X, X')

(Theorem 1.3.4.2 a) ), and so

j!E O j E ' -- 1E' b) By a),

.! u o jE' = jE' o 2E o jE' = j E ' ,

U O U _ _ U O j E , o j'E - - j E , O J"E - - U .

Hence u is a projection in E'" and it follows from Im jE' = Im (u o jE') C Im u = Im (jE' o J~E) C Im jE' ,

Im u = Im jE' that u is a projection of E"' onto I m j E , . By Theorem 1.3.6.3, IIJEI]-< 1,

so that

IIJE'II _< 1,

218

1. Banach Spaces

llull = IIJE' o J~ll ~ llJE'II llJ~ll ~

1

(Corollary 1.2.1.5, Theorem 1.3.4.2 b)). c) By a), jl

"I

"I

"I

E 0 U -- #E 0 jE' 0 #E "-- 3E"

Hence Ker j~ ~ Keru = Ker (jE' o J~E) ~ Ker j ~ ,

Ker u -- Ker j~ - (ImjE) ~

(Proposition 1.3.5.8). d) follows from b), c) and Murray's Theorem. e) The maps (E"/ImjE)'

(ImjE) ~

> (ImjE) ~

~ E ' " / I m jE, ,

x' ~ ~, q'x' ,

x'" ,

~ rx'"

are isometries (d), Proposition 1.3.5.2, Proposition 1.2.5.2 a =a c), and the assertion now follows, m

Corollary 1.3.6.20 ( 3 )

Let ~T be the Stone-Cech compactification of

the discrete space T . Put A :=

and take each x C e ~ ( T ) co(T)", and co(T)'"

ZT\T,

to be extended continuously to ~ T . Identify co(T)',

with g l ( T ) , / ~ ( T ) ,

and the Banach space M(flT)

Radon measures on ~ T , respectively (Example 1.3.6.2). Let i and j

of

denote

the evaluation of co(T) and /~I(T), respectively, and put u:=joi'. Then i and j

are the inclusion maps, u is the projection of M ( f l T )

gl(T) , Kern is the Banach space M ( A )

onto

of Radon measures on A ,

M(flT) = el(T)@ M(A), and e ~ ( T ) / c o ( T )

and ( e ~ ( T ) / c o ( T ) ) ' are canonically isometric to C(A) and

.hl ( A ) , respectively,

m

1.3 The Hahn-Banach Theorem

P r o p o s i t i o n 1.3.6.21

( 0 )

Let E

vector subspace of E ' . Put G := (~ map. Put H "= E l ~

be a Banach space and F

219

a closed

~ and let j " F --~ G be the inclusion

and let q" E ~ H be the quotient map. Finally let u

be the isometry H'

, ~ G,

~' ,

> q'~'

(Corollary 1.3.5.5). \ ~" ~ "11

b)

\ \ J .l-] " l " ~ ]

.

.

.

.

.

.

(jg~) o u -1 is continuous on GE whenever ~ E H .

c) If F # is dense in G#E, then II(JH~) o u - l l Fll = II~]l

for every ~ E H .

d)

If F # is dense in G#E and if every continuous linear f o r m on F

is

continuous on FE, then the map v" H

> F' ,

~ ~ } (jH~) o u -1 I F

is an isometry and uov' ojF = j.

e)

Under the hypothesies of d), if we identified F" with G via the isometry u o v' then j becomes the evaluation of F .

a) We have ((jHqx) 0 u - l , rl} = (jHqx, u-lrl) = (qx, u-lrl) = (x, q'u-lrl) = (x, r]) .

b) follows from a). c) Since F # is dense in GE~ , it follows from b) that I[(JH~) o u-1 I Fll = Ii(JH~)o u - l I =

IIJH(II-

I[~ll

(Theorem 1.3.6.3 a) ). d) Let ~' E F ' . By the hypothesis of d) and Corollary 1.2.6.5, there is an x E E such that

220

1. Banach Spaces

~'(~) = (x,~)

for every ~ E F . By a), (jHqx) o U-1 I F = ~'.

Hence v is surjective. By c), it is an isometry. Take ( E F and x E E . Bya), (x, uv'jF~) = (x, q'v'jF~} = (vqx, jF~) = (vqx, ~) = ( ( j s q x ) o U-1, ~) -- (X, ~) .

We deduce that uV'jF~ = ~ = j ( ,

u o v'ojF -- j .

e) follows from d) (and Corollary 1.3.4.7). P r o p o s i t i o n 1.3.6.22

I

( 1 ) Let T be a set and ~ S the Stone-Cech com-

pactification of the infinite discrete space S . Put

Take k E

~..o,1(S, T ) , and let k be the continuous extention of the map S

)~I(T),

Zs

s,

~ k(s,.)

~ e '(T)i',(~),,

where gl(T) is identified with a subspace of gl(T)" via the evaluation map.

~) k e e~

~?f k l ~ = O.

b) k E go'l(S, T) iff k ( A ) C co(T) ~ , where co(T) is identified with a subspace of gl(T)' via the evaluation map (and Example 1.2.2.3 e) ). The continuous extention k exists by the Alaoglu-Bourbaki Theorem. a) Let ~ be the filter on S consisting of all cofinite subsets of S, i.e. 9-- { A E ~ ( S ) ] S \ A

k lA -- 0 is equivalent to

finite}.

1.3 The Hahn-Banach Theorem

221

x' e el(T)' = : , lim(k(s,-) x ' ) = O. By Corollary 1.2.3.13, this is equivalent to lim Ilk(s, )Ill = 0 i.e. to k c t~~

T).

b) Take k e go'I(S,T) and s e A. Then (e T k ( s ) } =

lim (e T , k ( r , - ) ) -

lim k ( r , t ) - O

S ~ r --. s

S ~ r-+s

for every t C T , where t~I(T)' is canonically identified with t ~ ( T ) (Example 1.2.2.3 d)), so that k(s) e (IK(T)) ~ . Since ]K (T) is dense in co(T) (Proposition 1.1.2.6 c)), k(s) e co(T) ~ . Hence k(A) C co(T) ~ Now suppose that k(A) C co(T) ~ . Then lim k(r, t) =

lim (e T, k(r,-)}

S ~r--~ s

--

( e Tt ,

k(8))

--

0

S~r---~ s

for every s E A and t E T . Hence k(.,t) E co(S) for every t c T . Thus k e eo'~(S,T).

I

E x a m p l e 1.3.6.23 ( 1 ) Let S , T be sets and I~S the Stone-Cech compactifieation of S with respect to the discrete topology on S . Put A := ~ s \ s and for k C g~'~(S, T) let "k denote the continuous extension of the map

s

>e~(T),

s,

>k(s,.)

to

A-

~ ~o(T)~

with the identifications in Proposition 1.3.6.22. Put

M .- {klk ~ eT'I(S,T)},

U ' eo'~ ( S, T )

". M ,

k .

~~ ,

and endow .M with the norm

M.

> IR+,

k,

> supllk(s)ll. sCA

222

1. Banach Spaces

Then Ker u -- 60'1(S, T)

and the factorization

~.~"(S,T)/e~ of u through g o ' I ( S , T ) / g ~

>M

is an isometry.

By Proposition 1.3.6.22 a), Ker u = t'~ (S, T ) . By L e m m a 1.2.4.6, the factorization

v" e ~ ' I ( S , T ) / ~ ~ of u through g o ' I ( S , T ) / g ~

~ .A4

is bijective. Take K E g o ' I ( S , T ) / g ~

Then

il~Kli = ikl] = ~up ]lk(~)ii _< sup ilk(~,-)ii~ = i kii sEA sES for every k C K , so t h a t

II~gli < Ilgi]. Take a E IR with a < []vKl], and k E K . There is an s C A , such that IIk(~)ll > - .

Furthermore, there is an x' e (t'l(T)') # with

J<x', k(~))l

> ~.

Then lim I(x', k(r, .))] = I(x', k(s))] > a .

Sgr--~s

Hence there is an r E S such t h a t

< I<x',k(r, ")>I < Ilk(r, ")ll, < IlkllThus

< IIKII and JivKiJ ~ IIKII, since k and a are arbitrary. Hence ilvKiI = [IKII.

I

1.3 The Hahn-Banach Theorem

P r o p o s i t i o n 1.3.6.24

223

( 4 ) Let T be a compact space and # a positive

R a d o n measure on T

with support equal to T . D e n o t e by u and v the eva-

luation map of C(T)

and L ~ ( # ) ,

C(T)--+ L ~ 1 7 6

respectively, and by w the inclusion map

Then

(Im v) M (Im w") = Im(w" o u) = Im(v o w). Take a e ( I m v ) ~ (Imw"). There are x e L~176 and x" e C ( T ) " such that a - vx

wttx tt .

--

By the Vitali-Lusin Theorem, there is a disjoint sequence (Kn)ne~ of compact sets of T such that for every n C IN, x I K n is continuous and K,~ is the support of e g n ' # and such that

#(T\ U Kn)= o. n E IN

Let to E T and suppose x} U Kn has two distinct limits

OL1

and a2 in

nEIN

to. Put [0~ 1 - - 0/2[ C "-----

Take k C {1,2}. Put

s

"- {U (3 Ak [ U open neighbourhood of to},

denote by ~k an ultrafilter on Hk finer than the lower section filter of Hk (ordered by inclusion), and define x'k " L ~ ( # )

'

) IK

) lA,ak im ~ 1

y,

Then x'k E L ~ ( # ) ' and

(y, ~ ' x l ) - (~y, 4 ) = y(t0) for every y E C ( T ) . Hence I

W

I

X 1 ~

!

W

!

X 2

9

J(A yd#.

223

1. Banach Spaces

Since (x, x') = (vx, x'} - (w"x", x') = (x", w'x') for every x' E L ~ ( # ) ' , we get

(X, X'I) -- (X, X;). But

I(x,x'k)

~kl

-

= lim 1 A,~k ~

fA(X _ ak)d# < c

for every k c {1, 2} and we get the contradictory relation 3e = lal

-

a~l _ I~, - ( ~ , ~ i ) l + I ( x , G ) - ~ 1 _ 2c.

Hence x[ U Kn has a unique limit at to. nc1N Since to is arbitrary, we may extend x[ U Kn continuously on T . Thus nEIN xEImw, aCIm(vow),and (Im v) n (Im w") C Im (v o w). By Proposition 1.3.6.16, w" o u = v o w, so that Im(v o w) = Im(w" o u), and this implies Im(v o w) C (Im v) n (Im w"). Proposition

1.3.6.25

( 7 )

m

Let E , F be n o r m e d spaces and u C s

Then

(Im jE') N (Im u'") = Im(jE, o u') = Im (u'" o jF')Take x'" E I m ( j E , ) n (Im u'"). There are x' c E' and y'" C F " such that x m =jE,

X ~ =umy

m.

By Proposition 1.3.6.19 a) and Proposition 1.3.6.16 (and Corollary 1.3.4.5), x'

- - J "'" EJE, X

'

~- J "' E u

" 'y" '

--

(u"

0

jE)'Y"'

7_ ( j R

0

u)'y"

- - U ' 3"F Y

"'

,

xm -- jE 'xl = jE' u I'l 2FY ,t e jE' o u ' ( F ' ) = u ' " o j r , ( F ' ) .

Hence (Im jE') n (Im u") c Im(jE, o u') -- Im(u"' o j r , ) . The reverse inclusion is trivial.

II

1.3 The Hahn-Banach Theorem

Proposition

225

Let E be a complex Banach space and F' a vector

1.3.6.26

subspace of E' such that EF, is Hausdorff. We denote by E the underlying N

real Banach space of E and by G' the set of continuous linear forms on EF, and put y' " E

~,

x :

~ y'(x) - iy'(ix)

for every y' E G'. a)

G' is a vector subspace of E ' , y' c F' for every y' E G', the map G'

> F' ,

y' l

> y'

is an isometry of real normed spaces (with respect to the induced norms), and F'

> G'

x' t

>rex'

is its inverse. b)

If F ' = E' then G ' = E ' .

c)

/ f t h e map

E - - - + F",

z,

~ (z,.)lF'

is an isometry of complex Banach spaces then the map N

E

~a",

z,

>(z,.)la'

is an isometry of real Banach spaces. N

a) It is obvious t h a t G' is a vector subspace of E ' . By L e m m a 1.3.1.5, y' is a linear form on E and by L e m m a 1.2.6.4, y' C F ' . Moreover, N

r e y ' ( x ) = re (y'(x) - iy'(ix)) = y ' ( x ) , for every x C E , so t h a t re y' = y'. For x ' E F '

and x C E , N

r e x ' ( x ) -- (x, rex') - i(ix, rex') --

226

1. Banach Spaces

= re (x, x'} - i re i(x, x'> = re(x, x') + i ira<x, x') - (x, x'), so that re x' = x'. Hence the two given maps are bijective and everyone of them is the inverse of the other map. Since they are obviously IR-linear, it folllows from Corollary 1.2.1.6, that they are isometries of real normed spaces. b) follows from a) and Proposition 1.3.6.12. c) Let y" E G". Put x"'F'

>~,

x',

~y"(rex')-iy"(irex').

By a), Lemma 1.3.1.5, and Corollary 1.2.1.6, x" e F" and ]lx"ll = IlY"]I-By the hypothesis of c), there is an x E E such that

(x,.)lF' = x",

I xll -I1~"11- Ily"ll.

Take y' E G'. By a), y' C F ' and (x,y') = (x",y') = y " ( r e y ' ) - i y " ( i r e y ' ) .

Since (x, y') = (x, y ; - iy'(ix)

it follows

(z, y') - r Since y' is arbitrary

(x, .)IV' = r Hence the map

E---+

G",

z,

) (z,-)lG'

is an isometry of real Banach spaces. P r o p o s i t i o n 1.3.6.27 E' and

( 0 )

u-E

B

Let E be a Banach space, F a subspace of

)F',

z,

~(z,.)lF.

If EeF is compact, then u is an isometry of Banach spaces.

1.3 The Hahn-Banach Theorem

227

The map EF

} FtF,

xl

)'l,tx

being continuous, u ( E #) is a compact and therefore a closed set of F ~ . For every y E F , Ilyll = sup I ( y , x ) l = xEE#

~up I ( y , ~ x ) l

xEE#

By Corollary 1.3.1.9, u ( E #) = F '# . Since EF# is compact, EF is Hausdorffand so u is injective. Since EF# is compact, EF is Hausdorff and so u is injective. It follows that u is an isometry of Banach spaces.

I

228

1. Banach Spaces

1.3.7 T h e K r e i n - S m u l i a n T h e o r e m /

P r o p o s i t i o n 1.3.7.1

(

0

x

)

Let E

be a normed space and K

a weakly

compact convex set of E . Then

{ ~ I(,~, ~)e z # • K} is a weakly compact, absolute convex set of E containing K .

Put u : IK • EE,

~, EE, ,

(a,x),

~ax.

Then u is continuous, so that {ax I ( a , x ) 9 IK# x K} = u(IK # x K) is weakly compact. This set is obviously absolutely convex and contains K . I T h e o r e m 1.3.7.2

( 0 ) (Mackey) Let E be a normed space, ~ the set of

convex weakly compact sets of E , and A' a convex set of E' which is closed with respect to the topology on E I of uniform convergence on J~. Then A' is a closed set of ErE.

Take x' 9 E ' \ A ' . There is a K 9 ~l, such that {y' 9 E' [ x 9 K ==~ I ( x , x ' - Y'}I <- 1} C E ' \ A ' .

Put K' := {y' E E ' i x

C K =::v [x'(x)l < 1}.

Then K ~ is an absolutely convex set so that x' e E ' \ ( A ' + K ' ) . Every y' C E' is bounded on K . Thus there is an c~ C 1R+ with y' E a K ' . By Proposition 1.3.1.6, there is a linear form x" on E' which is bounded on K' such that sup rex"(y') < rex"(x'). y' cA'

We show that x" belongs to I m j E . Put

1.3 The Hahn-Banach Theorem

229

a := sup Ix"(y')l. y'EK'

We may assume that x" :/= 0, so that a =/: 0. P u t L := {~x I (/~,x) E IK # x g } . By Proposition 1.3.7.1, L is absolutely convex, weakly compact and contains K . Assume t h a t ~x" r j E ( L ) , jE(L) is an absolutely convex, compact set of E~,. By Proposition 1.3.1.8, there is a y' E E ~ such that /~ "- sup re (jEx, y') < re I x"(y'). xE L

O~

Since jE(L) is absolutely convex, /3 = sup [(jEX, y'}[ = sup ](x, y')[. xEL 1 ! E In particular, ~y

K !

xEL

and

a >_ Ix"(/jy')l >

rex"(y') > 5 c ~ -

o~

which is a contradiction. Hence ~1 x l t E jE(L) and there is an x E E with X" :

jE(X).

Then

sup re y' (x) -- sup re x" (y') < re x" (x') -- re x' (x). y~ E A ~

y~ E A ~

Therefore x' is not in the closure in E~ of A' and so A' is a closed set of E ~ . I Theorem

1.3.7.3

( 0 )

(Krein-Smulian, 1940) Let E be a Banach space

and A' a convex set of E ' . If A ' M n E '# is a closed set of ErE for every n E IN, then A' is a closed set of E~E. Let %' be the finest topology on E t inducing the topology of pointwise convergence on the equicontinuous sets of E ~ (Theorem 1.2.8.2). Since every equicontinuous set of E' is contained in a set of the form n E '# (n E IN), E ' \ A ' E ~s i.e. A' is closed with respect to %'. By Corollary 1.2.8.3, A' is closed with respect to the topology on E ~ of uniform convergence on the convex compact sets of E and the assertion follows from Theorem 1.3.7.2 since every compact set is weakly compact. I

Remark.

The theorem no longer holds if E is not complete.

230

1. Banach Spaces

D e f i n i t i o n 1.3.7.4

( 0 )

Let E

be a vector space. A cone of E

is a

nonempty subset A of E for which a A c A whenever a E JR+. The cone A is called s h a r p if

A M ( - A ) = {0}. 0 belongs to every cone. The cone A is convex iff A+ACA.

C o r o l l a r y 1.3.7.5

( 0 ) Let E be a Banach space and A' a convex cone

of E ' . Then A' is a closed set of ErE iff A' A E '# is a closed set of E~E .

First assume that A' N E ~# is a closed set of E ~ . Take n C IN and put

1,. ?2

Since u is continuous and A' A n E '# - u 1(A' A E ' # ) , A ' A n E '# is a closed set of E ~ . By the Krein-Smulian Theorem, A' is a closed

set of E ~ . The reverse implication follows from Proposition 1.2.6.6. Proposition

1.3.7.6

( 0 )

Let E

I

be a Banach space and (u~),e, a finite

family of projections in E' such that Im u, is a closed set of E~E and that UL o U x - - 0

for distinct ~, A E I . Then ~

Im u~

is a closed set of E~E .

tCI

First observe that ~ u~ is a projection in E' and that

Im E u ~

= EImu~.

LEI

~EI

Let x' be point of adherence of E '# A ~ Im u~ in E ~ . There is an ultrafilter LEI

;~ on E' converging to x' in E~ with E ~# N E I m u ~ LEI

E ~.

1.3 The Hahn-Banach Theorem

231

By the Alaoglu-Bourbaki Theorem, x' E E '# and ut(~) converges in E~ for every ~ E I . Moreover, lim u~(~) E I m u~, since Im ut is a closed set of E~. Then lim (~-'~ ut(~)) = ~ tEI

lim ut(~) E ~

tEI

tel

Imu~.

tEI

tel

x ' = lim ~ - l i m (:~-~ u ~ ) ( ~ ) = lim (~--~ ut(~')) E ~ tEI

tEI

Imu~.

tEI

Hence E '# N ~ Imut is a closed set of E~. By Corollaryl.3.7.5, ~ Im ut is tel

tel

a closed set of E~.

I

P r o p o s i t i o n 1.3.7.7

( 0 ) Let E be a normed space. Given a linear f o r m

x" on E ' , the following are equivalent:

a) x" E I m j E . b)

Kerx" is a closed set of E'E.

c) x" is continuous on E'E. a :=~ b and a ==~ c are trivial. b ==>a. We may assume that x" =/=0. Take x' E E'\Ker x". By Proposition 1.2.6.2, there is a finite subset A of E such that {y' E E' l x E A ~

](x,x'-

y')] < 1} C E ' \ K e r x " .

Then x" is bounded on {y' E E' i x E A ==~ I(x,y'>[ < 1}

as can be seen by factorizing x" through E'/Ker x". Hence, by Lemma 1.2.6.4, there is an x E E with x " ' - - (x,.)

and x" E ImjE. C ::=>a. Put

232

1. Banach Spaces

V' := {x' 9 E' I Ix"(x')l < 1}. By Proposition 1.2.6.2, there is a finite subset A of E such t h a t

Ix'(x)l <

{x' e E' l x e A ~

1} C V'.

By Lemma 1.2.6.4, x" is a linear combination of the ((x,-}),cA and so it belongs to I m j E .

I

C o r o l l a r y 1.3.7.8

Let E , F

be normed spaces. Given u C s

the

following are equivalent: a)

There is a v 9 s

b)

The map

with u = v'.

E'E

>F) ,

x' ,

> ux'

is continuous. c)

u'(ImjF) C ImjE. a =~ b. Given (x',y) c E' x F ,

= (y, v'x'> = (vy, ~') (Theorem 1.3.4.2 a) ) and the assertion now follows. b =:> c. Take y E F . Then (u'jFy, x') = (jpy, ux'} = (y, ux'> for every x' E E' (Theorem 1.3.4.2 a) ). Thus the map

Ek

~IK, z',

~(u3ry, z') !

9

is continuous. By Proposition 1.3.7.7 c =~ a, u'jFy belongs to I m j E . Hence u'(Im jR) C Im j E . c ~ a. Put v :F ~ (Corollary 1.3.6.5). Then v E s

E,

y,

) j~xlu'jry

E) and

(y, v'x'} = (vy, x'} = (u'jFy, x'} = (jpy, u x ' } -

(y, ux')

for every (x', y) C E' x F (Theorem 1.3.4.2 a)) and so u = v'.

I

1.3 The Hahn-Banach Theorem

/

Proposition

1.3.7.9

(

0

233

\

) Let E be a Banach space. Given a linear f o r m

x" on E ' , the following are equivalent: The restriction of x" to E '# is continuous at 0 with respect to the topology on E '# of uniform convergence on the weakly compact convex sets orE.

b)

E ' # N K e r x " is a closed subset of ErE .

c)

K e r x " is a closed set o.f E ~ .

d)

x" E I m j E . a ==>b. Let ~ be the topology on E ~ of uniform convergence on the weakly

compact convex sets of E and |

the topology induced on E '# by %7'. Take

x ~ E E '# and c > 0. There is a weakly compact convex set K of E such t h a t

I~"(y')l <

C

for every y' C E '# n K ' , where K ' : - {z' e E ' l x

Iz'(x)[ ~_ 1}.

c K ~

E ~# N (x' + 2 K ~) is a neighbourhood of x' in E ~# with respect to |

Take

y' C E '# N (x' + 2 K ' ) . Then

~1 ( y ' - x') e

E, #

n

K! '

SO

Hence the restriction of x" to E r# is continuous at x' with respect to | x' is arbitrary, x " l E '# is continuous with respect to | closed with respect to |

Since

Hence E '# NKer x" is

Since E '# is closed with respect to ~7' (Proposition

1.2.6.6), E ~# n K e r x " is also closed with respect to ~s

Being a convex set,

E '# n K e r x " is a closed set of E~ (Theorem 1.3.7.2).

b =v c follows from Corollary 1.3.7.5. a ~ d follows from Proposition 1.3.7.7 b ::v a. d =v a is trivial,

m

234

1. Banach Spaces

[

D e f i n i t i o n 1.3.7.10

(

0

) Let E, F be vector spaces. The map u" E --+ F

is called c o n j u g a t e - l i n e a r if u(crx + fly) = -Sux + / 3 u y for every x , y E E and a , ~ C IK.

Proposition

1.3.7.11

( 0 )

Let E , F

be Banach spaces. Given a linear

(resp. conjugate-linear) map u" E' -+ F ' , the following are equivalent:

a)

The map E'E

~ F~F ,

x' ~ ~ ux'

E#E,

) F'F,

x' ~ > ux'

is continuous.

b)

Themap

is continuous at O.

a :=> b is trivial. b =:v a. Take y C F . By b), the map EE#, ----+ IK ,

x' :

; (y, ux')

(resp. (y, ux'> )

is continuous at 0. By Proposition 1.3.7.9, a =:v d, there is an x C E such that (y, ux') = (x, x')

(resp. (y, ux') - (x, x') )

for every x' E E ' . a) now follows. [

C o r o l l a r y 1.3.7.12

~

0

) Let E be a Banach space. Given a projection u

in E ' , the following are equivalent:

a)

The map

is continuous.

b)

I

I m u and Keru are closed sets of E~E .

1.3 The Hahn-Banach Theorem

235

Put v := 1E --U. Then Im u = Ker v. a =~ b. Ker u is obviously a closed set of E ~ . The map

E'E

~, E'E,

x"~

~, vx'

is continuous, so that Ker v is a closed set of E ~ . Hence I m u is a closed set of EE,. b =~ a. Let ~ be an ultrafilter on EE#, converging to 0. Let x' denote the limit of u(~) in EE#, (Alaoglu-Bourbaki Theorem). Then v(~) converges to - x ' in EE#,. By b), x'CImu, and

-x'EImv=Keru

so

X~ -- ux ~ = O. Hence the map

E#E,

~ E'E,

x"

~, ux'

is continuous at 0. By Proposition 1.3.7.11 b :=> a, the map

is continuous. Lemma

1.3.7.13

I

Let T be a compact space. Take x E C(T) , and let (Xn)neIN

be a sequence in C(T) for which x is a point of adherence in the topology of pointwise convergence. If every subsequence of (x,~)ne~ has a point of adherence in C(T) with respect to the topology of pointwise convergence, then there is a subsequence of (xn)nc~

converging to x in the topology of pointwise conver-

gence. First assume that T is separable. Then, by the diagonal procedure, we may construct a subsequence (Yn)ne~ of (Xn)ne~ converging to x on a dense set

236

1. Banach Spaces

of T . Take t E T . Assume that (Yn(t))nC~ does not converge to x(t). Then there is an e > 0 together with a subsequence (z,~)ne~ of (Yn),~e~N, such that

Iz~(t) - x(t) l > for every n E IN. By assumption, (z,)nc~ has a point of adherence z in C(T) with respect to the topology of pointwise convergence. Then

Iz(t)-

~(t)l > ~,

and this is a contradiction, since z and x coincide on an dense set of T . Hence (Yn(t))nEIN convergens to x(t). Since t is arbitrary, (yn)ne~ converges to x in the topology of pointwise convergence. Now let T be arbitrary and let S:= H

xn(T),

nEIN

~'T

>S,

r

t,

~a(T),

> (x~(t))~e~,

t,

)~(t),

and 7rp'~a(T)

~IK,

(Sn)neIN'

;Sp

for every p E IN. Then p is continuous and ~a(T) is compact. Let y be a point of adherence of (Xn)nE~ in C(T) with respect to the topology of pointwise convergence, and take t', t" E T such that

r

- r

Then

xn(t')=xn(t") for every n E IN, so that

y(t') = y(t"). Hence there is a unique map y: S ~ IK with y=yor

1.3 The Hahn-Banach Theorem

Since p(T) is the quotient space of T with respect to ~ ,

237

y is continuous, x is

a point of adherence of (Tr~)ne~ and every subsequence of (Trn)ne~ has a point of adherence in C(~(T)) with respect to the topology of pointwise convergence. Since p(T) is separable, the first part of the proof implies the existence of a subsequence (Trk~)ne~ of (Trn)ne~ converging to x in the topology of pointwise convergence. It follows that the subsequence (xkn)~e~ of (x~)ne~ converges to x in the topology of pointwise convergence.

I

L e m m a 1.3.7.14 Let T be a compact space and let C(T)T denote the set C(T) endowed with the topology of pointwise convergence. Given 9c C C ( T ) ,

the following are equivalent: a)

Every sequence in Y has a point of adherence in C(T)T.

b)

Every sequence in T contains a sequence which converges in C(T)T.

c)

jc is a relatively compact set of C(T)T. b :=~ a and c :::> a are trivial. a =:~ b follows from L e m m a 1.3.7.13. a :=> c. Let ~ be an ultrafilter on C(T) containing IF. By a), {x(t)lx e ~ }

is bounded for every t E T , so t h a t the map x'T

>IK,

t,

> l i m y(t )

is well-defined. We show that x is continuous. Take t E T and c > 0. Assume t h a t every neighbourhood of t contains a point s such that

I x ( ~ ) - ~(t)l > ~ We construct inductively a sequence (tn)neli in T starting with tl "= t and a sequence (X~)ne~ in 9c such t h a t the following hold for every n C IN" 1) n--/: 1 ~

]X(tn)- x(t)] > e.

2)

I x ( t k ) - x~(tk)l < ~1 for every k C ]N~_I

3)

Ixk(tn)- xk(t)l < n1

for every k C INn

Choose Xl arbitrarily. Take n C IN, n > 1, and assume that the sequences have been constructed up to n -

1. By the definition of x , there is an xn C 9c

such that 2) is fulfilled. Since the functions in 9r are continuous,

238

1. Banach Spaces

n

~ T I I x k ( s ) - xk(t)l < 1 } k:l

is a neighbourhood of t. By hypothesis, there is a tn in this neighbourhood of T satisfying 1). This finishes the inductive construction. Let s be a point of adherence of (tn),~e~ and y a point of adherence of

(Xn)n~IN in C(T)T. Then

y(tk) - x(tk) for every k C IN, by 2). Thus

l y ( ~ ) - x(t)l > by 1). By 3),

xk(~) = x~(t) for every k E IN, so that

y(~) - y(t) = y(t~) = x(t~) - ~ ( t ) , which is a contradiction. Hence there is a neighbourhood V of t such that

for every s E V, and so x is continuous at t. It follows that x r C(T), converges to x in C(T)T, and Y is relatively compact. T h e o r e m 1.3.7.15

9

Let A be a subset of the Banach space E . Then the

following are equivalent: a)

Every sequence in A has a point of weak adherence in E .

b)

Every sequence in A has a weakly convergent subsequence in E .

c)

A is weakly relatively compact. By the Alaoglu-Bourbaki Theorem, EE#, is compact. Given x C E , define

~:'E~, and

,~IK,

x',

~(z,x')

1.3 The Hahn-Banach Theorem

~. E---~C(E~,), a ~

~-~

239

~.

b. Let (xn)ne~ be a sequence in A. By a), every subsequence of

(~(xn))ne~ has a point of adherence in C(E~#,) with respect to the topology of pointwise convergence. By Lemma 1.3.7.13, there is a strictly increasing sequence (k~)n~iN in 1N such that (~(Xkn)),~eIN converges to some y 9 C(E#E,) in the topology of pointwise convergence. Define x ' " E'

~ IK,

x', ~ " lim (x~n x').

Then x" is linear and Xll l E t #

-_ y .

Hence E '# N Ker x" is a closed set of E ~ . By Proposition 1.3.7.9 b =:~ d, there is an x C E , such that jEX = X".

Then (Xk.)neIN converges weakly to x. a =:v c. Let ~" be an ultrafilter on E containing d . By a), {x'(x) Ix e A} is a bounded set of IK for every x' c E ' . Thus the map x 'r" E' ---+ IK,

x' ~

limx'(x) x,~

is well-defined. It is obviously linear. By a), every sequence in ~p(A) has a point of adherence in C(E~#,) with respect to the topology of pointwise convergence. By Lemma 1.3.7.14 a ::~ c, ~(~) converges to some y e C(E#E,) with respect to the topology of pointwise convergence. We have x'r lE '# = y ,

so E '# N Ker x" is a closed set of E ~ . By Proposition 1.3.7.9 b ~ d, there is an x E E with jEX -- X".

Then ~ converges weakly to x , and A is weakly relatively compact. b :=> a and c ==> a are trivial.

I

Remark. a) The implication a ~ b was proved by Smulian (1940) and the implication a =~ c was proved by Eberlein (1947).

b) It is possible to prove a stronger form of Lemma 1.3.7.14 (for T a compact instead of compact) so that the above theorem can be proved without the use of Proposition 1.3.7.9.

240

1. Banach Spaces

1.3.8 Reflexive Spaces Definition 1.3.8.1 ( 0 ) (H. Hahn, 1927) A normed space is called reflexive if its evaluation map is surjective (in which case it is an isometry (Corollary 1.3.6.5)). It may happen that a Banach space is isometric to its bidual without being reflexive (R.C. James, A non-reflexive Banach space isometric with its second conjugate space, Proc. Nat. Acad. Sci. USA 37 (1958) 174-177).

Proposition 1.3.8.2

( 0 ) Every finite-dimensional normed space is retie-

xive.

This follows immediately from the fact that the dual and the algebraic dual of a finite dimensioual normed space coincide (Corollary 1.2.4.10). 9 P r o p o s i t i o n 1.3.8.3 ( 0 ) Every reflexive space is complete and its bounded sets are weakly relatively compact. Let E be a reflexive space. Then, using the evaluation map, we may identify E with E" and EE#, with ~"# By Corollary 1 2.1.10 E is complete and, by the Alaoglu-Bourbaki Theorem, EE#, is compact. Hence every bounded set of E is weakly relatively compact. 9 E

Proposition 1.3.8.4

I

9

.

,

( 0 ) (P.J. Pettis, 1938) A Banach space E is retie-

zive iff E' is reflexive.

If E is reflexive, then E' is obviously reflexive. Assume that E is not reflexive. Identify E with a subspace of E" via the evaluation map. Since E is complete, it is a closed subspace of E". By Corollary 1.3.3.6, there is an x'" E E'"\{0} vanishing on E . Then x'" does not belong to ImjE, and so E' is not reflexive. II P r o p o s i t i o n 1.3.8.5

Let E be a normed space and F a subspace of E .

a) F is reflexive iff j E ( F ) = F ~176 b)

If E is reflexive and F is closed, then F is reflexive (P.J. Pettis, 1938).

1.3 The H a h n - B a n a c h Theorem

241

a) Let j 9F ~ E be the inclusion map. T h e n Im j "

--

F ~176

(Proposition 1.3.6.17). If F is reflexive, then F ~176 = Im j " = Im (j" o j F ) = Im (jE o j ) -- j E ( F ) (Proposition 1.3.6.16). Now suppose t h a t j E ( F ) -- F ~176 . Take y" E F " . T h e n j " y " C F ~176 , and so there is an x C F with jEX =

j"y".

Then .,,. 2 3FX

__

jEjX =jEjx

__

j,,y,,

(Proposition 1.3.6.16) and y" C j F x

(Proposition 1.3.6.17). Hence jF is surjective and F is reflexive. b) Take x" E F ~176 . Since E is reflexive, there is an x C E with x" = jEX .

Given x ~ c F ~ ,

(x, x'> = (jEX,X'> = (X", X'> = 0 so t h a t m

xE~176 (Proposition 1.3.5.7). Hence F ~176 C jE(F).

The reverse inclusion is trivial. By a), F is reflexive. Proposition of E ,

1.3.8.6

Let E

be a n o r m e d space, F

and q" E " --~ E ' / F ~176the quotient map.

I a closed vector subspace

242

1. Banach Spaces

a)

ElF

b)

If E is reflexive, then so is E / F .

is reflexive iff q o jE is surjective.

a) Let r : E --+ E l F

be the quotient map. Then jE/F o r = r" o jE

(Proposition 1.3.6.16), so that jE/F is surjective iff r" o jE is surjective. By Proposition 1.3.6.18, the factorization of r" through E " / F ~176is an isometry. Thus r" o jE is surjective iff q o jE is surjective. b) If E is reflexive, then jE is surjective. Thus q o jE is surjective. By a), E/F

is reflexive.

C o r o l l a r y 1.3.8.7

I Let F be a closed subspace of the normed space E . Then

E is reflexive iff F and E l F

are reflexive.

The necessity follows from Proposition 1.3.8.5 b) and Proposition 1.3.8.6 b). For the converse, assume that F and E l F

are both reflexive and let

q: E" -+ E " / F ~176 be the quotient map. Take x" C E " . By Proposition 1.3.8.6

a), there is an x C E such that qjEX = qx".

Then q(x" - jEX) = O,

x" - jEX E F ~176 By Proposition 1.3.8.5 a), there is a y with jEY -- X tt -- jEX.

Hence x" = j ~ ( x + y),

i.e. jE is surjective and so E is reflexive. P r o p o s i t i o n 1.3.8.8

I

( 0 ) A Banach space, which is isomorphic to a refle-

xive Banach space, is itself reflexive.

1.3 The Hahn-Banach Theorem

243

Let u : E --+ F be an isomorphism of Banach spaces and assume E reflexive. Then u" is surjective (Corollary 1.3.4.7). Since

jF o u

-

-

~tt 0 jE

(Proposition 1.3.6.16), it follows that j F o u is surjective. Hence jF is surjective and F is reflexive. E x a m p l e 1.3.8.9

every set T .

I ( ]. ) ( 7 )

g P ( T ) i s reflexive for every p r

o0[ and

c0(T), c(T), gX(T) and g~(T) are reflexive iff T is finite.

For gP(T) (p e [1, c~] U {0}) this follows from Example 1.2.2.3 d),e) and Proposition 1.3.8.4. By Example 1.2.2.4 c), c(T) and co(T) are isomorphic, so that the assertion for c(T) follows from that for co(T) and from Proposition 1.3.8.8.

I

Example 1.3.8.10 Let S, T be sets and p, q E ]l, cx)[ be conjugate. Then gP,q(S, T) is reflexive. This is an immediate consequence of Example 1.3.8.9 and Proposition 1.2.3.6 b). Examp'r6')l.3.8.11

I

If It is a measure and p e ]1, oc[, then LP(It) is reflexive.

The assertion follows from Example 1.2.2.5 c).

I

E x a m p l e 1.3.8.12 If T is a completely regular space, then C(T) is reflexive

iff T is finite. If T is finite, then C(T) is reflexive py Proposition 1.3.8.2. Assume that T is infinite. Replacing T by its Stone-(~ech compactification, if necessary, we may assume that T is compact. There is a sequence (tn)ne~ in T for which

tn ~ {tin I m e ]N\{n}} for every n E IN. First suppose that there are two distinct ultrafilters ~ , ~ on IN with lim tn = lim tn. n,~ n,~ Take A r ~'\$ and put

B:={tnlneA}.

244

1. Banach Spaces

Then

eB

is a Borel function on T and

#J ) f eB d#

IK ,

x"" 2t4b(Z)

is a continuous linear form on Adb. Take x 9 C(T). Then limx(tn) - l i m x ( t n ) n,~

n,~

'

and so there is an n 9 IN with

9(t~) # ~ ( t ~ ) . Hence

jC(T)X r X". Thus jC(T) is not surjective (Example 1.2.2.10), and C(T) is not reflexive. Now suppose that for any two distinct ultrafilters ;~, ~ on IN lim t , r lim t , . n,;~

n,~

Then {tin I n 9 IN} is homeomorphic to the Stone-0ech compactification of IN. Define 9v := {x e C(T) in 9 IN ==, x(tn) = 0},

u" C(T) ~

~oo,

x,

> (x(tn))

nEIN

Then Ker u = 9c and the factorization of u through C(T)/5 c is an isometry (Tietze's Theorem). By Example 1.3.8.9, go~ is not reflexive, so that C(T)/.~ is not reflexive. By Proposition 1.3.8.6 b), C(T) is not reflexive.

I

Every bounded sequence in a reflexive Banach space has a weakly convergent subsequence. P r o p o s i t i o n 1.3.8.13

By Proposition 1.3.8.3, every bounded set of a reflexive space is weakly relatively compact and the assertion now follows from Theorem 1.3.7.15 c =:> b.

I

1.3 The Hahn-Banach Theorem

1.3.9 C o m p l e t i o n

of N o r m e d [

D e f i n i t i o n 1.3.9.1

(

0

245

Spaces

) Let E be a normed space. A c o m p l e t i o n o f E is

a Banach space F such that E is a dense subspace of F . Theorem ImjE

1.3.9.2 ( 0 )

Let E

be a normed space. I f E

(Corollary 1.3.6.5), then I m j E [

Theorem

1.3.9.3

(

0

is identified with

is a completion of E .

ll

\

) Let E be a normed space and let F , G be comple-

tions of E . Then there is a unique isometry u" F --+ G with U X z X

for every x E E . The uniqueness of u is trivial. Let jl " E -+ F ,

j2 " E ~

G be the

inclusion maps. Then, by Proposition 1.2.1.13, we can extend them to operators j-~ C E.(G, F ) , j~ E / : ( F , G ) , respectively, with IlJl I = IIJ21 = 1. We have jl o j2(x) - x ,

j2 o jl (x) = x

for every x C E . We deduce ~oj-~=lF,

j-2ojl= la.

u " - j 2 now has the required properties. Remark.

The above theorem allows us to identify all completions of E . This

justifies the use of the term the completion of E .

246

1. Banach Spaces

1.3.10 A n a l y t i c Functions Definition 1.3.10.1

( 0 ) Let E be a Banach space and U an open set

of IK. A function f

9U -+ E is called analytic if for given so C U there is

a power series ~ t n x n i n E

and an r > 0 such that r is smaller than the

n--O

radius of convergence of this power series,

U~(ao) c u, and oo

:(,) = E(,-,o)o,. n----0

for every s C U~IK(C~o).

By Proposition 1.1.6.11, if f " U ~ E and g" U ~ E are analytic, then s f + ~g" U ~ E is analytic for all s,/~ E IK. P r o p o s i t i o n 1.3.10.2

Analytic functions are differentiable and their deriva-

tives ~re analytic.

The proposition follows immediately from Proposition 1.1.6.25. oo

P r o p o s i t i o n 1.3.10.3

( 0 ) Let E be a Banach space, ~ n--O

series in E ,

r its radius of convergence, So C IK, and oo

f " Urn(SO)

>E ,

s,

> ~-~.(s - So)" X, . n--O

Then f is analytic.

Take ~0 E U~(s0) and put r'=~-I~0

- Z01

Take a E UT,(/~0) and let p : = I~ - ~ol + 19o - ~ol < ~' + I~o - ~ol = ~ .

Then

(:),o m--0

Oonm

/o

Oo, n

I

t'x. a power

1.3 The Hahn-Banach Theorem

247

for n E IN U { 0 } , and so

i[(n)

,o

for n C IN U {0} and m e IN. U { 0 } . Hence oo

oo,n

E

.

, ( cx:).

Given m C IN U { 0 } , put

ym := ~

(~o - ~o)"-mx.

n---m

(Corollary 1.1.6.10 a =~ c) and

(:) ::o for n C IN with m > n . Take p C IN. Then

I~- ~olmllYmll -< ~ m=0

: E

I~o -~ol~-mltz~l

m=0

[]Xn[[

]ZO- OzO[n-mloL- ~0[ m

n=0

I~- 9o:m =

n=0

~ E

m=0

]]Xn]lpn < 00.

n=0 [

(Corollary 1.1.6.10). Since p is arbitrary, the family

[(a-

\

~O)mym)meiNU{O } -

is absolutely summable. It follows that the radius of convergence of the power (DO

series ~

tmym is greater than r' (Theorem 1.1.6.23).

m--0

Take c > 0. There is a p C IN such that

f(~)-~-~(~-~o

x~ < 5 ,

~

n=0

I1~11 < 5

m=p+l

Then (Proposition 1.1.6.11, Corollary 1.1.6.10),

E

(OL -- /~O)mym -- E m=0

m=0

=E n--0

(OL -- /~0)m

(2~0 -- OLO)n-mxn n=0

( ~ - ~o/~(~0- ~0)o-~ ~~ m--0

--

248

1. Banach Spaces

=

(~ - ~ o ) - x . +

Z

n=O

n=p+l

(~ - ~o)m(9o - ~o) --m

Xn,

m=O

P

P

II ~ - ~ ( ~ - ~o)mym- ~-~(~- ~o)'x~ll _< m=0

_< ~ n=p+l

n=O

I~.11

I~-~o1~1~o-~ol "-~

< ~

m=0

I~.11o'<~,

n=p+l

o0

p

Ilf(~) - ~

(~ - :3o)~Ymll ~ I f(~) -- ~

n=0

(~ - ~0)~x~ll+

n=O

P

+ll Z

P

(" - "0) nz` - Z

(" - Z~

+

m=0

n--O

oo E

E

E

m=p+ 1

Since c is arbitrary, oo

f (a) - ~

(a - ao)mYm .

m=O

Since a and /3o are arbitrary, f is analytic. P r o p o s i t i o n 1.3.10.4

( 0 ) Let E , F be Banach spaces. Take u E s

Let U be an open set of IK and f " U - + E an analytic function. Then u o f is analytic. oo

Take a0 E U. There is a power series ~

tnxn in E and an r > 0, such

n=0

that r is smaller than the radius of convergence of the power series,

u~(ao) c u, and (2O

:(-)- E(---o)'-o n--O

1.3 The Hahn-Banach Theorem

249

for every a E Ur~(ao). By Corollary 1.2.1.17, r is smaller than the radius of oo

convergence of the power series ~

tnux~ and

n--0 oo

n--0

for every a E Ufi(ao). Hence u o f is analytic. C o r o l l a r y 1.3.10.5

I

Let E be a Banach space, U a domain in ]K, and

f : U -+ E an analytic function. If f vanishes on an open nonempty subset of U, then f vanishes identically.

Take x ~ E E ' . By Proposition 1.3.10.4, x' o f is analytic. Since it vanishes on an open nonempty subset of U, it vanishes identically. By Corollary 1.3.3.8 a), f vanishes identically. T h e o r e m 1.3.10.6

I

( 0 ) (Liouville's Theorem) Let E be a complex Banach

space. Every bounded analytic function (~ --+ E is constant.

Let f :~ -+ E be a bounded analytic function and take a,/~ E ff~. Assume that f (a) ~ f (/3). Then there is an x' E E' with X' o f (a) ~: X' o f (fl)

(Corollary 1.3.3.9). By Proposition 1.3.10.4, x t o f is analytic. Since it is bounded, it is constant by the classical form of Liouville's Theorem. Hence x' o f (a) = x' o f (j3)

which is a contradiction. Remark.

I

The above theorem was proved by Cauchy (1844) for E = ~ .

C o r o l l a r y 1.3.10.7

I

~

0

\

) Let E be a complex Banach space and f "r

an analytic function. If

lim f(o~) = 0 t~----~o o

then f is identically zero.

E

250

1. Banach Spaces

f is bounded, so it is constant (Theorem 1.3.10.6).

I

T h e o r e m 1.3.10.8 ( 6 ) (Laurent's Theorem, 1843) Let E be a complex Banach space. Take so E 9 and 0 < rl < r2. Put V : - {c~ E e l r l

< Ic~-sol < r2},

and let f " U -+ E be an analytic function. Then there is a unique family (xn)ne~ in E such that oo

n'----O0

for every a E U. The radius of convergence of the power series Orb

O0

Et"Xn

(resp.

n:0

Et~x_n) n:l

is greater than r2 (resp. ~1 ) . The expression oo

(t -

;

n'----O0

is called the Laurent series of f , -1 E

(t-

OLo)nXn

n:--O0

is called its principal part and x-1 is called its residue.

Take r E ]rl, r2[. Given n E IN, put

1/

2rr

xn := 27rrn

f(a0 + reit)e-intdt,

0

where the integral is defined (as in the classical case) with the help of the Riemann sums. Take n E IN. Then 27r

Xn , X I) - -

1 f 27crn

x ' o f(c~o + reit)e-i'adt

0

for every x' E E ' . By Cauchy's Theorem, (zn, x') does not depend on r. Hence, by Corollary 1.3.3.9, xn does not depend on r.

1.3 The Hahn-Banach Theorem

Set fl "= sup IIf(ao + ~e")ll < ~ . tEIR

Then

I(x,~, x')l <_ IIx'lls7 r n

for n E ~

and x ~ E E ~.Hence

IIx~ll_ ~ for every n E IN (Corollary 1.3.3.8 b) ). Thus lim sup lixnii! <_ 1 n--~(x)

r

lim sup I x _ . II~- <

~.

n----~ o o

Since r is arbitrary, the radius of convergence of (x)

Zt~

oo

(,esp ~t"x_,,I

n--O

n"-I

1 is greater than r2 (resp. K)"

Take a E U. By the classical Laurent's Theorem, CX3

x, o s ( ~ ) =

E/xo,~'/(~-~o) n----

~

(x)

for x ~ E E ~. Hence oo

~s(<~/,x')-~ E (<~- ~o)nxo, lrl,--"

--

x ')

CX:)

for every x' E E' (Corollary 1.2.1.17) and so oo

n----oo

(Corollary 1.3.3.9). To prove the uniqueness, let (Xn)neZa be a family in E , such that oo

n----- oo

251

252

1. Banach Spaces

for every c~ C U. Choose x' C E'. Then oo

x,o

E

o

'n---- oo

(Corollary 1.2.1.17). Then 2~"

( Z n , X t) ---

2~r

2 7 [1r n /

xt o f(c~o+reit)e-intdt= ( 27TF 1--~-n f

0

0

f (ao + reit)e-intdt, x ')

for every n C 2Z by the Theorem of Residues. Since x' is arbitrary, 27r

x,~ = 27rrn

f (~o + reit)e-intdt 0

for every n E 2Z (Corollary1.3.3.9). Remark.

In the proof we have used only the fact that f is continuous and

x' o f is analytic for every x' E E ' .

Definition 1.3.10.9 ( 6 ) Let E be a complex Banach space, U an open set of ~ , ao c U, f " U \ { a o } ~ E an analytic function, and r > O such that

c u The Laurent series of fI(U~ (c~0)\{c~0}) is called the Laurent series of f in C~o and its principal part (residue) the principal part (the residue) of f

at

C~o. If there is a p E IN, such that the principal part of f at C~o has the form -p

Z

(t-

rt----1

with X_p ~ O, then we say that f has a pole at (~o o.f order p .

By Laurent's Theorem, the principal part of f at c~0 and the residue of f at a0 are fully determined by f , and the principal part of f is convergent for every c~ C ~\{c~0}. In particular, if c~0 is a pole of f , then its order is determined by f . P r o p o s i t i o n 1.3.10.10 ( 0 ) Let E , F , G be Banach spaces, u ' E • a continuous bilinear map, U an open set of IK, and f, g analytic functions on U with values in E and F , respectively. Then the map U is also analytic.

)G,

a,

~u(f(oz),g(o~))

1.3 The Hahn-Banach Theorem

(DO

Take s0 c U. There are power series ~

253

O0

tnxn, ~

rt=0

t~y~ in E and F ,

n=0

respectively, and an r > 0 such that r is smaller than the radii of convergence of these power series,

Ur~(aO) C U, and

(DO f (c~) - ~

(DO

(~ - C~o)nX~, g(a) - ~

n--0

(c~ -

OLo)nyn

n--0

for every a e U~(ao). Given n e IN tO {0}, set n

:= Z

u(x ,

.

k--0

By Proposition 1.2.9.6 and Theorem 1.1.6.2:3, the radius of convergence of the oo

power series ~

tnzn is greater than r and

n=-0

E (OL- OLo)nZn--?Jr( E (OL- OLo)nXn,E (0~- OLo)nyn)~- u(f(oL), g(oL)) n=0

n=0

n=0

for every a E U~(ao). Hence the map U is analytic,

>G,

a,

>u(/(a).g(a)) m

254

1. Banach Spaces

Exercises E 1.3.1

Let E be a real vector space and p a real function on E satisfying

p(x + y) <_ p(~) + p(y) ,

p(~)

= ,w(~)

for every x , y 9 E and a > 0. Show t h a t

~) p(0) = o. b)

p ( - x ) > - p ( x ) for every x 9 E .

E 1.3.2

Let E be a normed space. Take a > 0 and let (Xn)ner~ be a sequence

in E . Let (a~)ne~ be a sequence in IK. Prove t h a t the following are equivalent: a)

There is an x' e E ' , such that IIx'll ~ ~ and

x'(x.) = ~ n for every n E IN.

•

b)

O~k/~k


•

/~kxk

k=l

k=l

for every n E IN and (flk)ke~ E IK n . E 1.3.3

(Phillips, 1940) Let T be a set, E a normed space and F a subspace

of E . Take u C s

g ~ ( T ) ) . Show that there is a v C / : ( E , g ~ ( T ) ) , such t h a t

~lP = ~, E 1.3.4 a)

Let G be a normed space. Show that the following are equivalent:

Given a subspace F of the normed space E , and u C s

a v Cs b)

I1~11 = Ilvll

G), there is

with vlF = u.

If E is a normed space, then every subspace of E which is isomorphic to G is a complemented subspace of E .

c)

Given a set T , every subspace of g ~ ( T ) which is isometric to G is a complemented subspace of g ~ ( T ) .

1.3 The Hahn-Banach Theorem

255

Show further that G is complete whenever the above conditions are fulfilled. (Hint: Use E 1.3.3 and E 1.2.15 to prove c =~ a.) E 1.3.5

(Kottman) Show that every infinite-dimensional normed space con-

tains a sequence (xn)ne~ such that

IIx~ll = 1,

II~-

~fl > 1

for distinct m, n E IN. (This is an improvement on Corollary 1.1.4.3.) E 1.3.6

Take p E [1, cx~[. Let q be the conjugate expontent of p, and ur, u~,

respectively, the right and left shift on t~P(23). Show that u r' and ut' are the left and right shift of gq(23), respectively. E 1.3.7

Take n E I N . L e t

K be a compact set of ]Kn . G i V e n

P E l K [ S l , . . . , s~, t l , . . . , tn], let P be the map K Show that { P I P E 1.3.8

> 1K,

a ~ ~P(al,...,

E IK[sx,..., sn, t l , . . . , t n ] }

an, ~ 1 , - . . , ~ n ) 9 is a dense set of C ( K ) .

Let (T~)~z be a finite family of compact spaces and for (x~)~i C

1-I C (T~) define | x~ " I-I T ~

t6I

> IK ,

t,

tCI

>l-I~(t~) t6I

Show th~t {~, x~ I (x~)~, e 1-I C(Z)} is ~ d~nse s~t of C(H Z)tCI

E 1.3.9

Let E , F

E and F , respectively. Take u C s a)

u(G) ~

b)

u'(H ~ C ul(H) ~ .

c)

The inclusion in b) may be strict.

E 1.3.10

tel

be normed spaces. Let G and H be vector subspaces of F ) . Show:

u~(G~

Let ] K - ( g ,

U an open set of ]K, and

E "= {f e ~~

] / is analytic}.

Prove the following: a)

E is a closed vector subspace of g ~ ( U ) .

b)

E is complete with respect to the normed induced form g ~ ( U ) .

c)

The above assertions not longer hold when C is replaced by 1R.

256

1.4

i. Banach Spaces

Applications

of Baire's

Theorem

Baire's Theorem has two very important consequences in the theory of Banach spaces, namely the Principle of Uniform Boundedness is bounded whenever it is pointwise bounded Operators

a set of operators

and the Principle of Inverse

the inverse function to a bijective operator is continuous. The

latter principle, which was proved by Banach, is deep and probably the most frequently quoted theorem in this book. This is because invertible operators play a central role in the theory and the Principle of Inverse Operators provides a remarkably simple criterion for determining whether an operator is invertible: an operator is invertible in the category of Banach spaces if and only if it is invertible in the category of vector spaces.

1.4.1 T h e B a n a c h - S t e i n h a u s T h e o r e m P r o p o s i t i o n 1.4.1.1

( 0 ) Let E be a Banach space and A an absolutely

convex closed set of E such that E-

U (nA). n E IN

Then 0 is an interior point of A .

By Baire's Theorem, there is an n E IN such that n A has nonempty interior. Hence there are x E E and c > 0 with U~(x) C A .

Since A is absolutely convex,

U~(-x) C

A.

It follows that

y - ~ l(x- + y ) +

1 x+y) ~(-

EA

for every y E U~(0), i.e. U~(0) C A and 0 is an interior point of A. T h e o r e m 1.4.1.2

I

( 0 ) (Banach-Steinhaus) Let E be a Banach space, F

a normed space, and .T" a set of s

F ) , such that

sup I1~11 < uE.~

for every x E E . Then ~

is bounded.

1.4 Applications of Baire's Theorem

257

Put

A :: N ~ ( F # ) uE.F

A is an absolutely convex closed set of E

(Proposition 1.2.7.2, Proposition

1.2.7.7, Proposition 1.2.7.4). Take x 6 E . Then

~up II~ll _< ~, uE.F

for some n C IN. Then sn

C A and x 6 n A . Hence

U (~A) - E nEIN

By Proposition 1.4.1.1, 0 is an interior point of A , i.e. U~(0) c A, for some e > 0. Thus ex E A and

II~xll- ~1 I1~(~) I _< ~ for every u C 3c and x C E # . Hence 1

11~11 _< -g

for every u 6 ~" (Proposition 1.2.1.4 b)), i.e. 9r is bounded. Remark.

1

The above theorem is occasionally called the Principle of Uniform

Boundedness. It was proved in 1923 by Banach for sequences of operators and in 1927 by Banach and Steinhaus in the above form. The condition t h a t E be complete cannot be relaxed (see Exercise 1.4.1). C o r o l l a r y 1.4.1.3 (un)~cIN in s

( 0

)

Let E , F

be Banach spaces. Take a sequence

F) such that (UnX)neIN converges for every x C E and con-

sider u:E Then u 6 s

>F,

x,

> limunx. n---~oo

F), Ilull _< lim inf ]lunl] _< sup Ilu.I < oc, n--+oo

and (Un)nEIN c o n v e r g e s

nEIN

to u uniformly on every compact set of E .

258

1. Banach Spaces

By the Banach-Steinhaus Theorem, sup II~.ll <

~

nEIN

The other assertions follow from Proposition 1.2.1.7 and Proposition 1.1.2.15.

m C o r o l l a r y 1.4.1.4

( 0 ) Let E be a normed space and A a weakly bounded

set of E , i.e. sup Ix'(x)l < xEA

.for every x' E E ' . Then A is bounded. In particular, the weakly compact sets of E are bounded.

Given x' E E ' , supl<jEx, x')l = sup I(x,x')l < oo. xEA

xEA

By the Banach-Steinhaus Theorem, j E ( A ) is a bounded set of E". Hence A is a bounded set of E (Theorem 1.3.6.3 a)). I Remark.

This corollary was proved for sequences bv Hahn in 1922.

C o r o l l a r y 1.4.1.5 ( 0 ) Let E be a Banach space, F a normed space, and take 9v C s F ) . If sup I(ux, y'}l < oo uE~

for every (x, y') E E • F ' , then ~ is bounded.

By Corollary 1.4.1.4, { u x i u E $'} is bounded whenever x E E . Thus, by the Banach-Steinhaus Theorem, $" is bounded. I C o r o l l a r y 1.4.1.6

( 0 ) Let u" E ~ F be a linear map between normed

spaces. Then the following are equivalent:

a) u is continuous. b) y, o u is continuous for every y' E F ' . c) u is continuous with respect to the weak topologies on E and F .

1.4 Applications of Baire's Theorem

259

a =:~ b is trivial. b =~ c. Take x E E and let ~ be a filter on E converging weakly to x . Finally, take y' E F ' . By b) and Proposition 1.3.6.12, y' o u is weakly continuous and so y'(u(~)) converges to y ' ( u x ) . Since y' is arbitrary, u(~) converges weakly to ux and u is continuous with respect to the weak topologies on E and F . c =~ a. Take y' C F ' . By Proposition 1.3.6.12, y' is weakly continuous. Hence, by c), y ' o u is weakly continuous too. Using Proposition 1.3.6.12, again, we see that y~ o u is continuous and so

s~p ly'(~x)l = ~up ly' o~(~)l < xCE#

xEE#

(Proposition 1.2.1.1 a ~ e). By Corollary 1.4.1.4, u ( E #) is bounded, so t h a t u is continuous (Proposition 1.2.1.1 e =~ a). Corollary 1.4.1.7

m

( 0 ) Let E be a vector space and p,q norms on E . Let

Ep (rasp. E q ) be the vector space E endowed with the norm p (rasp. q ), and p' (rasp. q') be the canonical norm on (E~)' (rasp. (Eq)'). Then the following are equivalent: a)

p < (~q for some ~ C ]1%+.

b)

(E~)' c (E~)' a~d q'l(E,)' <__Zp' fo~ ~ o ~ ~ e ~§

r

(E~)'c (Eq)'.

In particular, p and q are equivalent iff the vector spaces (Ep)', (E q)' coincide. a ~ b. Let j : Eq --+ Ep be the identity m a p on E . By a) (and Proposition 1.2.1.1 d) :=~ a), j is continuous, j' : (Ep)' -+ (Eq)' is the inclusion map and the existence of /3 follows from Proposition 1.2.1.1 a =~ d. b =~ c is trivial. c =~ a follows from Corollary 1.4.1.6 b =~ a (and Proposition 1.2.1.1 a =~

d). The final assertion follows from a r C o r o l l a r y 1.4.1.8

Let E , F , G

c.

be Banach spaces and u : E • F -+ G a

bilinear map. If u is continuous in each variable, then u is continuous. Given y E F ,

sup lu(~, y)t _< sup I1~(-, y)ll IIxll = II~(-, y)ll < xEE#

xEE#

m

260

1. Banach Spaces

(Proposition 1.2.1.4 a)). By the Banach-Steinhaus Theorem c~ :-= sup Ilu(~, )ll < oo. xEE#

Take (x, y) (5 E x F with x -7/=0. Then ~lXllxll(5 E # , so that

ti~(x, y)it = ,txi, ti~(~x, ~),. _< ,Ixii ilu(ijx~X,.)ll llyii _< ~tixtl tiyl, (Proposition 1.2.1.4 b)). By Proposition 1.2.9.2 c ==> a, u is continuous, Proposition

1.4.1.9

i

(Gelfand) Let A be a subset of the separable Banach

space E . Then the following are equivalent. a)

A is relatively compact.

b)

Every sequence in E' which converges pointwise to 0 converges uniformly to 0 on A . a ==> b follows from Corollary 1.4.1.3. b => a. We first show that A is bounded. By Corollary 1.4.1.4, it is sufficient

to show that A is weakly bounded, i.e. that sup Ix'(x) l < oo xEA

for every x' (5 E ' . Assume the contrary. Then there is a sequence (xn)net~ in A and an x' (5 E' such that

I*'(*~) I > for every n E IN. Then (g1 Xt)nEI N is a sequence in E' which converges pointwise to 0. But it does not converge to 0 uniformly on A, which contradicts b). Hence A is bounded. Let ( x ~ ) n ~ be a dense sequence in E and put

B:= {~ I~e~}. Then E ' ~

is a compact space (Alaoglu-Bourbaki Theorem and Proposition

1.1.2.15). Set u" E

>C ( E ' # ) ,

x,

~ jEX

i E'#.

u preserves norms (Corollary 1.3.3.8 b), Theorem 1.3.6.3 a)) and so I m u is a closed set of C(E'#B) (since E is complete). In order to show that A is

1.4 Applications of Baire's Theorem

261

relatively compact, we must therefore prove that u(A) is a relatively compact set of C(E'#B). Since u(A) is a bounded set of C(E'#u), it is sufficient to show that u(A) is equicontinuous (Ascoli Theorem). Take x' E E'#B and c > 0. We show that there is a neighbourhood U of x t in E~B # such that

I(x, y')- (~, ~')l < for every yt E U and x E A. Assume the contrary and take n E IN. Then 1 n

is a neighbourhood of x' in E ' ~ . There are yn E A and x~ E Un such that !

I(y~, ~ -

x')l _> c.

Then lim X'n(X~ ) = X' (xk)

n--+ oo

for every k E IN and so (X~z)nEIN converges to x' in E'p#. By Proposition !

1.1.2.15, (x~)ns~ converges to x' in E~ and by b), (Xn)nE~ converges to x' uniformly on A. Hence e _< lim I(Yn, x'n

_

X r

n--+(x)

)1 = 0 I

and this is a contradicition. E x a m p l e 1.4.1.10

(4)

Let Sl "-- {OL E r

] [OL[- 1}

and 27r

XnA .__ ~1 f

e_in tx(t)dt

0

for

X E C(Sl)

and n E 2Z. There is an x E C(S1) such that

p E IN

does not converge, i.e. the Fourier series of x is not pointwise convergent.

269

1. Banach Spaces

Given p E IN, set P n=-p

and P X p! " C ( S 1 )

) ]Z ,

X l

) E

Xn 9

n---p

For p E IN and t El0, 27r[

fp(eit ) Xp'

= sin(p + sin t

1)t

is linear and 27r

x,,(~)

1 /

=

fp(eit)x(t)dt.

o

Thus

1/ [fp(dt)ldt ~ Ixll~ ~ 27r

14r

o

for every x E C(S1) 9Hence xp' is c o n t i n u o u s and

1/

27r

Ix;l[ <_ ~

Ifv(eit)l dt

0

( P r o p o s i t i o n 1.2.1.1 d => a). Define

f "S1

) ]K

Take g > O. There is an x E

1/

'

a'

C(S,)

fv(e it)

1

>

-1

if f p ( a ) > 0 if f p ( a ) < 0.

such t h a t ]lxlloo _ 1 and

(xlet) - .lezt))dt < e .

o

Then

I

x'p(x) - ~1

2/

]fp (eit )[dt
0

1.4 Applications of Baire's Theorem

263

and

IIx'p[[>_[x~p(x)l>_-~1/ [fp(e~t)ldt- c 27r

0

(Proposition 1.2.1.4 b)). Since ~ is arbitrary, 27r

[IXpll > -

1 [ ]fp(eit)[dt > ~

27r

1/

,sin(P+ 89

"It

o

t

dt =

o

(2p+1)~ (k+l)~ 1 / ]sint[dt- 1~o / ]sin nt[ dt > t 7r t

7r

0

--

k~

(k+l)lr

1~-~.

- > -T"

k=O

1

(k + 1)7r

flsinntldt_2~-~l k~

~

k=O

k+l

Hence lim ][x;I I - oc.

p---~c~

By Corollary 1.4.1.3, there is an x e C(S1) for which converge,

(Xp(X))p~

does not m

264

1. Banach Spaces

1.4.2 Open Mapping Principle Proposition

1.4.2.1

( 0 )

Let E be a normed space, F a Banach space,

and u " E -+ F a linear surjective map. Then 0 is an interior point of u ( E # ) . u ( E # ) is absolutely convex (Propositions 1.2.7.2, 1.2.7.7, and 1.2.7.5), and U

nu(E#)D

n E IN

U

U

nu(E#)-

n E IN

u(nE#)-

n E IN

u(U

nE#):

By Proposition 1.4.1.1, 0 is an interior point of u ( E # ) . Proposition

1.4.2.2

and take u E s

( 0 )

u(E)-

F.

n E IN

I

Let E be a Banach space, F a normed space

F) . If 0 is an interior point of u ( E # ) , then 0 is an interior

point of u ( E # ) .

By hypothesis, there is an c > 0 with cF # C u(E#).

We prove t h a t F# C u(E#).

Take y E ~ F ~ . We construct inductively a sequence (xn)~e~ in E # such that

I

1 )

y - u

2--~xm kin-- 1

< mc

2n

for every n E IN. Take n E IN and suppose that x~,...,x,~_~ have been constructed. Then y-u

~ Xm

E

F # C 2--~

km = 1

Hence there is an x,~ E E # with

y - u

~xm

c - ~---~uxn < 2n+l '

\m=l

i.e.

y - u

2--;xm m--1

2n_}_ 1 -

1.4 Applications of Baire's Theorem

265

This completes the inductive construction. (~nXn) is an absolutely convergent sequence in E . Put nE1N

1 X-- Xn nE IN

(Corollary 1.1.6.10 a =~ c). Then Iixll _< E

lIIxni[ _< 1

nEIN

(Corollary 1.1.6.10) and y=

lim u

n --+c~

(~2~ 1 m=l

-~

xm

)

=uxEu.E#,()

Hence

F# C u(E #) and 0 is an interior point of u(E#).

I

T h e o r e m 1.4.2.3 ( 0 ) (Open Mapping Principle, Banach 1932) Every surjective operator between two Banach spaces is open, i.e. maps open sets into open sets. Let E, F be Banach spaces and u " E --+ F a surjective operator. Let U be an open set of E and y C u(U). Take x E U with

ux=y. Then x + e E e c U, for some e > 0. Hence

y + ~ ( E # ) = ~(z + ~E#) c ~(U). By Proposition 1.4.2.1 and 1.4.2.2, 0 is an interior point of u(E#). Hence y is an interior point of u(U) and u(U) is thus open. I C o r o l l a r y 1.4.2.4 ( 0 ) (Principle of Inverse Operators) Every bijective operator between Banach spaces is an isomorphism.

266

1. Banach Spaces

Let E, F be Banach spaces and u : E ~ F a bijective operator. Since u is open (Theorem 1.4.2.3), u -1 is continuous. Hence u is an isomorphism,

i

Let E , F be Banach spaces. An operator u : E --+ F is an isomorphism (isometry) iff u' is an isomorphism (isometry).

C o r o l l a r y 1.4.2.5

The necessity was proved in Corollary 1.3.4.7. So assume that u' is an isomorphism (isometry). By Corollary 1.3.4.7, u" is an isomorphism (isometry). Then

u"

o jE -- jF o u

(Proposition 1.3.6.16), so that jE

=

'U'/-I O j E o u .

Hence Ilxll = IIjExll = itu ' ' - ' o jF o u(x)l I <_ IIu"-' o JFII II~xll

([ixli = iljExil = ]ljguxl] = Iluzll) for every x 6 E (Theorem 1.3.6.3 a), Proposition 1.2.1.4 a)). Hence u is injective and I m u is closed (Proposition 1.2.1.18 a), c)). Since u' is injective, I m u is dense (Corollary 1.3.5.9). Thus u is surjective. By Corollary 1.4.2.4, u is an isomorphism (isometry).

i

Corollary 1.4.2.6 ( 0 ) Let F, G be closed subspaces of the Banach space E such that the map FxG

>E,

(x,y),

)x+y

is bijective. Then E=F| F , G being Banach spaces, F • G, is also a Banach space (Proposition 1.1.5.1). The map

F•

>E,

(x, y),

>x+y

is linear and continuous (Proposition 1.2.5.1) and so it is an isomorphism(Corollary 1.4.2.4).

i

1.4 Applications of Baire's Theorem

C o r o l l a r y 1.4.2.7

( 0 ) Let E, F be Banach spaces and take u E s

267

F).

If F / I m u is finite-dimensional, then there is a finite-dimensional vector subspace G of F such that F=G@Imu.

Let v be the factorization of u through E / K e r u , q" F --+ F / I m u the quotient map, and (Y~)~eI an algebraic basis of F / I m u. Given ~ E I , take y~ C Y~ and let G be the vector subspace of F generated by (Y~)~eI. Being finite-dimensional, G is a Banach space (Corollary 1.1.3.5), so that (E/Keru) • G

is also a Banach space (Theorem 1.2.4.2 e), Proposition 1.1.5.1). Define w'(E/Keru)

•

~F,

(x,y),

>vx+y.

Being the composition of the operators >(Imu) • G ,

(E/Keru) • G

(Imu) •

>F,

(x,y),

} (vx, y ) ,

(z,y)~-+z+y

(Propositions 1.2.4.7 and 1.2.5.1) w is itself an operator. Take y C F . There is a family (c~)~er in IK such that qY - E

(~ Y~ "

Since q(y - Z

-

- Z

~CI

= 0

LEI

it follows that y- ~c~y~ C Kerq- Imu = Imv. Take x E E / K e r u such that y-

~

~y~ - v x .

Then y = vz + ~ ~EI

a~y~- w(z, ~ tEI

Hence w is surjective. Take (x, y) c Ker w. We have successively

a~y~) C Im w.

268

1. Banach @aces

vx+y=O, qy = - q v x = O,

y=0,

vx=O, xzO. Hence w is injective. By the Principle of Inverse Operators w is an isomorphism and so Im u = Im v is closed. Since the m a p G

>F/Imu,

y,

>qy

is an isomorphism, it follows from Proposition 1.2.5.2 c =:> a, that F=G@Imu.

C o r o l l a r y 1.4.2.8

( 0 ) Let E , F

i

be B a n a c h spaces and u C s

The

following assertions are equivalent:

a)

I m u is closed and u is injective.

b)

u is lower bounded, i.e. there is an c~ > 0 such that

f o r every x C E .

a => b. The operator

v:E

>Imu,

x,

> ux

is bijective. By the Principle of Inverse Operators, it is therefore an isomorphism. Put

1 flv-~il

"

Then

IIxll = IIv-l(ux))ll _< IIv-lil Iluxll, 1

II~xll _> jlv_lllllxll = ~flxll for every x E E . b =~ a follows from Proposition 1.2.1.18 a),c).

1.4 Applications of Baire's Theorem

Proposition s

1.4.2.9

(

0

)

Let E , F

269

be Banach spaces and take u E

F) such that Im u is closed. Let j : K e r u --~ E be the inclusion map.

a)

The associated algebraic isomorphism of u is an isomorphism of Banach spaces.

b)

Im u' = (Ker u) ~ = Ker jr,

(Ker u) ~176 = Ker u " .

c)

The factorization E r / I m u' --+ (Ker u) r of jr through E r / I m u' is a isometry. a) follows from the Principle of Inverse Operators (and T h e o r e m 1.2.4.2 e)). b) Take x' C (Ker u) ~ . Let r : E --+ E / K e r u be the quotient map and

the associated algebraic isomorphism of u. By Proposition 1.3.5.2, there is a y' C ( E / K e r u ) ' such t h a t x t = rryt = yr o r . By a), ~-1 is continuous, so that y' o ~-1 belongs to ( I m u ) '

By the H a h n -

Banach Theorem, there is a continuous linear form z' on F which extends y~ o ~ -1 Then

(~,~'z') = (~,x,z') = ( ~ , y '

o ~-~) = ~'(~-~(~x))

= y'(~x) = ~'(~)

for every x E E , and so x ~ = u'z' C I m u ' ,

(Ker u) ~ C Im u'. Since the reverse inclusion follows from Proposition 1.3.5.8, we have that (Ker u) ~ = Im u',

(Ker u) ~176 = (Ira u') ~ = Ker u" (Proposition 1.3.5.8). The fact that (Ker u) ~ = Ker j' was proved in Theorem 1.3.5.12 b). c) follows from b) and Theorem 1.3.5.12 c).

I

270

1. Banach Spaces

C o r o l l a r y 1.4.2.10 Let E and F be B a n a c h spaces. Take u E s with I m u closed. Let q" E ~ E / K e r u , r 9F' ~ F ' / K e r u ' be the quotient maps and i" Im u --~ F , j " Im u' --~ E ' the inclusion maps. Let g" E/Ker u --+ Im u and u' " F ' / K e r u -+ Imu' be the algebraic i s o m o r p h i s m s associated to u and u ~, respectively. Define

v" ( E / K e r u ) '

> (Keru) ~

x',

> q'x'

(Proposition 1.3.5.2) and let

w" F ' / ( I m u) ~

~ (Im u)'

(Proposition 1.3.5.8) be the factorization of i' through F ' / ( I m u ) ~ .

a)

Keru'-(Imu)

~

Imu'=(Keru) ~ Keru-~

b) ~ and

~'

Imu=~

are i s o m o r p h i s m s of B a n a c h spaces.

c) v and w are isometries. B

d)

u'-vog'

ow.

a) By Proposition 1.3.5.8 K e r u ' = (Imu) ~

Keru=

~

Imu-

~

and, by Proposition 1.4.2.9 b), Im u' - (Ker u) ~ . b) By Proposition 1.4.2.9 a), g is an isomorphism of Banach spaces. Hence, by Corollary 1.3.4.7, the same holds for g~. c) follows from Proposition 1.3.5.2 and Theorem 1.3.5.12 c). d) Given x E E and y~ E F ' , (x, j v ~ ' w r y ' } - (x, q'~'i'y') -- (i~qx, y'} = (ux, y'} = {x, u'y'} - ( x , j - ~ r y ' ) , SO

j

ovo~'

owor

-

j

ou' or.

Since j is injective and r is surjective, VO~lOW=U l .

I

1.4 Applications of Baire's Theorem

( 0 )

Proposition 1.4.2.11 s

Let E , F

271

be Banach spaces and take

u E

F ) . If I m u ' is closed, then Im u is also closed. Define G := I m u ,

v" E

~ G,

x'

~ ux,

w:G

~F,

y~

>y.

Then v' is injective (Corollary 1.3.5.9) and w' is surjective (Theorem 1.3.5.12 a)). Since

~'(a') = v ' ( ~ ' ( F ' ) ) = ~'(F') (Corollary 1.3.4.5), v'(G') is closed. By Corollary 1.4.2.8 a ~

b, there is an

c~ > 0 with

II~'Fti > ~lly'li for every y' c (7'. Take y C c~G#\v(E#). Since v ( E # ) is absolutely convex there is a y' C G' such that

sup

z~v(E#)

I(z, v')l < (y, y')

(Corollary 1.3.1.7). Thus

~ily'll ~ liv'FII = sup I(~, ~'y')l = sup I(vx, v')l < (y, y') <_ ~llv'll xEE#

xEE#

which is a contradiction. Hence

~G # c v (E#) and 0 is an interior point of v ( E # ) . By Proposition 1.4.2.2, 0 is an interior point of v ( E # ) . Thus v is surjective. It follows that I m u = v(E) = G and so I m u is closed in F . Proposition

1.4.2.12

Let E , F

m

be Banach spaces and r

vector subspaces of E . Take u E s

the set of closed

and let q : E --+ E / K e r u

quotient map, v the factorization of u through E / K e r u

be the

and w the algebraic

isomorphism associated to u. Then the followsing are equivalent:

272

1. Banach Spaces

a)

u(G) is closed for every G 9 6 .

b)

q(G) is closed for every G 9 ~5 and v is bounded below.

c)

q(G) is closed for every G C 0~ and w is an isomorphism. a =, b . Take G c 6 . It follows from

q(a)-

~'(~(a))

t h a t q(G) is closed. Since Im v = u(E) it follows t h a t Im v is closed. Thus, by Corollary 1.4.2.8 a ~

b, v is bounded

below. b =:~ c. By Corollary 1.4.2.8 b =~ a, Im v is closed. Hence I m u is closed and by the Principle of Inverse O p e r a t o r s , w is an isomorphism. c ==~ a. Since w is an isomorphism, Im w is complete and u(G) = w(q(G))

I

is closed in F for every G c 6 . Proposition

s

1.4.2.13

(

0

Let E , F

)

be Banach spaces and take u E

F ) . Let p be a projection of E onto Ker u and q a projection of F onto

Im u. There is a v C s

E ) , such that

1E--VOU = p ,

uov=q,

voq=v.

Define G := Ker p and w:G

>Imu,

x a

>ux.

T h e n w is obviously linear and continuous. For x C Ker w, UX

--

WX

=

O~

so t h a t x = 0, i.e. w is injective. Take y E I m u . T h e n

1.4 A p p l i c a t i o n s of B a i r e ' s T h e o r e m

273

y = U X ~

for some x E E . T h e n x-px

E G,

w(x - px) = ux - upx = ux = y ,

i.e. w is surjective. Since I m u is closed ( T h e o r e m 1.2.5.8 b ::v a, P r o p o s i t i o n 1.2.5.2 a => b ) , it is complete. By the Principle of Inverse Operators, w is an isomorphism. Set v" F

> E,

y,

) w-l(qy).

T h e n v is linear and continuous and voq--v.

For x E E , x - px E G ,

u ( x - p x ) -- u x ,

(1E -- V o U)X -- X -- W - I ( q u x )

--- X

--

W--I(?.t(X

--

px))

--- X -- W-I(UX) --

-- X -- (X -- px)

-- px,

so t h a t 1E--VoU--p.

For y E F , (u o v)y

-- u(w-l(qy))

-- qy

and so uov--q.

Proposition

~)

1.4.2.14

T h e r e is a v E s is a c o m p l e m e n t e d

Let E, F

I

be B a n a c h

such that uov subspace of E.

spaces and u E s

F).

= 1F i f f u is s u r j e c t i v e a n d K e r u

274

1. B a n a c h Spaces

b)

T h e r e is a v 6 s

E)

s u c h that v o u = 1E i f f u is i n j e c t i v e a n d Im u

is a c o m p l e m e n t e d s u b s p a c e o f F .

a) Assume first t h a t u o v = 1F for some v 6 s

E ) . Then u is surjective

and v is injective. P u t p:-vou.

Then pop=

vouovou

= vou-p,

i.e. p is a projection in E . For x 6 E ,

(px = o) ~

( w x = o) r

(~

= o),

so t h a t Ker u = Ker p and Ker u is a complemented subspace of E by M u r r a y ' s Theorem (Theorem 1.2.5.8 b ~ a). The reverse implication follows from Proposition 1.4.2.13 and Murray's Theorem. b) Assume first t h a t y o u = 1E for some v 6 s

E ) . Then u is injective.

Put p:--uov.

Then pop=

u ov

ou

ov

=

u ov

=

p,

i.e. p is a projection in F . If y C I m u then there is an x C E with y--ux.

It follows y = ux = uvux - pux C Imp,

so t h a t Im u C I m p .

1.4 Applications of Baire's Theorem

275

The reverse inclusion is trivial, so that Imu = Imp and Im u is a complemented subspace of F (Theorem 1.2.5.8 b =~ a). The reverse implication follows from Proposition 1.4.2.13 and Murray's

I

Theorem. Proposition

1.4.2.15

Let E be a Banach space and p be a projection of E"

onto I m j E . Put F := K e r p and u:E

...... ; ( ~

a)

u is an operator and

b)

Given y' E ( ~

x,

~(jEz) I~

Ilull ~< 1.

there is an x E E such that ~x

= V,

I1~[I _< Ilpll lly'll.

c)

u is a isomorphism iff F is closed in E ~ , .

d)

I f F is closed in E~, and IlpII <- i , then u is an isometry.

a) u is linear and

II~xll--II(jEx)l~

~ ItJEII = llxll

for every x E E (Theorem 1.3.6.3 a)). b) By the H a h n - B a n a c h Theorem, there is an x" E E" such that

x" l~

= y',

IIx"ll- Ily'll.

Take x E E with jEX = p x " .

Then

9" - p ~ " e F t so that

for every x' E ~

and thus

(~176

276

1. Banach Spaces

u x = yr.

Moreover, Ilxll = IIJExll = IlPX"II ~ IlPll IIx"ll = IlPll IlY'II 9

c) Assume that u is an isomorphism and take x" C (~

~ There is an

x E E such that jEX = p x " .

Since x" - p x " C F ,

it follows that (ux, x') = (jEx, x') = (px", x') = (x", x') = 0

for every x' C ~

Hence u x = O,

x = O,

r=

x" E F ,

(~

and F is a closed subset of E~,. Now suppose that F is closed in E~,. Take x E Ker u. Then jzz

e (~

~= F

(Proposition 1.3.5.4) and so jEX = O,

X = 0

(Theorem 1.3.6.3 a)). Hence u is injective. By b), u is surjective. By the Principle of Inverse Operators, u is an isomorphism. d) follows from a), b), and c). C o r o l l a r y 1.4.2.16 a) b)

E

I

Given a B a n a c h space E , the f o l l o w i n g are equivalent:

is a dual space.

There is a projection p of E " onto I m j E such that IIPI]-< 1 and Kerp is closed in E ~ , .

1.4 Applications of Baire's Theorem

277

a => b follows from Proposition 1.3.6.19 b), c). b ==~ a. Let F := K e r p and take

u :E

~ (~

x,

~ (jEx) I ~

I

By Proposition 1.4.2.15 d) u is an isometry.

Remark.

Let T be a Hausdorff space. Let # r

0 be an atomfree Radon

measure on T . Let j be the evaluation map of L I ( # ) . Then there is a projection p of L I(#) '' onto I m j

such t h a t

IIpll-< 1. But L l(#) is not a dual space

(Example 1.3.1.15). This shows that we cannot omit the hypothesis "Kerp is closed in E~," from b). Proposition

1.4.2.17

Let E be a Banach space and F , G , H

closed vector

subspaces of E such that E=FOG,

H=F•H+GNH.

If q denotes the quotient map E --+ E / H ,

and if q ( F ) , q(G) are closed, then

E / H = q(F) @ q(G) . Define

~ : q(F) x q(G)

Step 1

>E / H ,

(X,Y) ,

~ is injective

Take (X, Y ) c q(F) x q(G) with

X+Y Take x E F , y C G

-0.

with

X -- qx,

Y = qy.

Then

q(x + y) = X + Y = 0, so that

x+yEH.

>X + Y .

278

1. Banach Spaces

By hypothesis, there are a C F N H and b E G N H with

a+b=x+y. We see that

a-x=y-bEFMG and therefore

a-x=y-b=O,

x=aEFNH,

y=bCGNH,

X = qa = O,

Y=qb=O.

Hence ~a is injective. Step 2

~ is surjective

Take z C E . There are x C F and y E G such t h a t

z=x+y. We get

qz = qx + qy = ~a(qx, qy) E Im ~a. Hence, ~a is surjective. Step 3

E / H -- q(F) 9 q(G)

By the first two steps and Corollary 1.4.2.6

E / H = q(F) | q(G) . Definition 1.4.2.18

( 0 ) Let X , Y

1

be sets and take f . X - - + r .

The set

{(x, f ( x ) ) i x C X }

is called the graph of f . If X, Y are topological spaces, Y Hausdorff, and f : X - - + then the graph of f is closed. If E , F

Y continuous,

are vector spaces and u : E ~ F is a

linear map, then the graph of u is a vector subspace of E • F .

1.4 Applications of Baire's Theorem

T h e o r e m 1.4.2.19 E,F

( 0 )

279

(The Closed Graph Theorem, Banach 1929) Let

be Banach spaces and u 9 E --~ F

a linear map. I f the graph of u is

closed, then u is continuous.

Let p, q be the projections of E • F onto E and F , respectively. E • F is a Banach space (Proposition 1.1.5.1) and p, q are continuous. Let G be the graph of u. G is a vector subspace of E • F . Being closed, it is a Banach space. Put 9=

pie.

Then v is bijective. By the Principle of Inverse Operators (Corollary 1.4.2.4), v -1 is continuous. From u=qov

-1

I

it follows that u is also continuous. C o r o l l a r y 1.4.2.20

Let E , F

be Banach spaces, u " E ~

F

a linear map

and f " F' --4 E' an arbitrary map such that

(~x, y') - (x, f(y')) for every (x,y~) C E • F ' . Then u is continuous and f - u ' . m

Let G be the graph of u and take ( x , y ) E G . Then there is a sequence ( ( x , ~ , y n ) ) n ~ in G converging to ( x , y ) . Take y ' e F ' . Then

for every n E lN. Thus (ux, y'} - (x, f ( y ' ) ) = l i m (xn, f ( y ' ) ) = l i m (yn, y') - (y, y') .

Since y' is arbitrary, ux=y

(Corollary 1.3.3.9). Hence (x,y) E G and the graph of u is closed. By the Closed Graph Theorem, u is continuous. By Theorem 1.3.4.2 a) f = u'.

I

280

1. Banach Spaces

Exercises E 1.4.1

Take p C [1, oc] and let IK (~) be endowed with the norm induced

by t~p. Given x c IK (IN) and n c IN, set s

nx(n)

if m = n

Xn " IN

0

if m=/=n

x,

>xn.

and u n : ] K (~)

>IK (~),

Prove the following: a) b)

un C / : ( I K (~)) for every n C IN. lim unx = 0 for every x E IK (~) n----~ oo

c)

sup Ilunll = oc (i.e. the conclusion of the B a n a c h - S t e i n h a u s Theorem does nEIN

not hold). E 1.4.2

Let E , F

be Banach spaces and u : E ~

F a surjective operator.

Show t h a t there is an c~ > 0 such t h a t

II~'y'll _> ~lly'll for every y' E F ' . E 1.4.3

Let E be a vector space and let p < q be complete norms on E .

Show t h a t p and q are equivalent.

1.5 Banach Categories

281

1.5 B a n a c h Categories The set of operators on a Banach space forms a Banach algebra and the whole of Chapter II is devoted to the study of such algebras. Unfortunately this theory cannot be applied to the case of operators between two different Banach spaces. The corresponding general theory is the theory of Banach categories, which is the subject of this section. Note that a Banach algebra is no more than a Banach category with precisely one object. This theory is not developed further in this book and so the reader may choose to omit this paragraph. 1.5.1 D e f i n i t i o n s

Definition 1.5.1.1

( 1 )

(2)(3)

A B~h

~y.t~m is aclass $2

and a map fit defined on $22 such that fit(E, F) is a Banach space for every E, F E $2 and

(E, F) --/: (G, H) ~

A(E, F) n A(G, H) = 0

for every E, F, G, H C F2 . We use the expressions "the Banach system (~Q,fit)" or "the Banach system ,4 over Y2" or, simply, "the Banach system M". The elements of [2 are called the objects o f the B a n a c h s y s t e m and the elements of fit(E, F) (for E, F C $2) are called the m o v p h i s m s o f the B a n a c h system. We put

.A(E) := .A(E, E) for E C $2 and E -~ F :r

E 4

A

F " v:=, x E A ( E , F )

for E, F E s Let A, B, C be Banach systems over the same class F2 . An

(A, B, C)-

multiplication is a map ~ defined on ~23 such that the following holds for every E, F, G c $2 :

~(E, F, G) is a bilinear map

A ( E , F) • U(F, a )

~ C(E, a) ,

such that

Ilyzll _< Ilzll Ilyll

(x, y) ,

~ y~

282

1. Banach Spaces

for all E -~ F Y+ G. A left (resp. right) multiplication on A over B is an (.A, B,.A) (resp. (]3, .A, .A))-multiplication. A unit for such a multiplication is a map 1, defined on $2, such that

1~ E B(E), 1E :/: 0 ==V IIIEII = 1, 1EX = X (resp. XlE = X ) for every E, F E $2 and F--~ E (resp. E-5+ F ). A left and a right multipliA

A

cation on .4 is called compatible if (ax)b = a(xb) for every E, F, G, H E $2 and E-~

F4

A

a-~

H.

An inner multiplication on fit is a left (or right) multiplication on ,4 over ,4 such that (~y)z = ~(yz)

for every E, F, G, H E $2 and E - ~ F--~ G - ~ H . An inner multiplication has a unit if tts left and right multiplications have units. Let ($2,.A), ([2, B) be Banach systems. A left and a right multiplication on ~4 over B are called simultaneously compatible with an inner multiplication on.Aft (xa)y = x(ay) for every E, F, G, H E $2 and E4

A

F4

B

G4

A

H.

A B a n a c h category (unital B a n a c h category) is a Banach system endowed with an inner multiplication (which has a unit). If $2 is a class of Banach spaces, then the map

(E, F ) ,

)s

defined on $22 is a Banach system (Theorem 1.2.1.9). It is a unital Banach category with the usual composition of the maps as multiplication (Corollary 1.2.1.5). We denote it by s or, simply, s

1.5 Banach Categories

Example 1.5.1.2

283

Take p C {0} U [1, cx3] and let q be the conjugate exponent

of p. Let 32 be a class of sets. Then the Banach system

(s, T) ,

~ e~,~(S, T)

on 32 with the multiplication defined in Proposition 1.2.3.5 is a Banach category. I

The claim follows from Proposition 1.2.3.5.

Proposition 1.5.1.3 ( 2 ) Let .A,B,C be Banach systems on the same class 32, ~a an (A, B, C)-multiplication, and take E, F, G E 32. F

~) ~(E, F, G) i~ contin~o~ b) If (XL)tEI is a summable family in A ( E , F) (in B(F, G) ), then

tel

for every x e B(F,G)

t~:_I

tel

tC I

(x E A ( E , F ) ) .

a) follows from Proposition 1.2.9.2 c =:~ a. b) follows from a) and Proposition 1.2.1.16.

I

Proposition 1.5.1.4

( 2 ) Every unital Banach category admits a unique unit for the left multiplication and a unique unit for the right multiplication and they coincide; we call it the unit of the Banach category.

Let (32,A) be a Banach category and 1 (resp. 1') be a unit of the left (resp. right) multiplication of A. Then 1E= 1EI~ = 1~ for every E C 32. Definition 1.5.1.5

I ( 1 ) ( 2 ) ( 3 ) Let (32, A ) b e a u n i t a I B a n a c h

category and take E, F E 32. An element x c A(E, F) is called left invertible (right invertible) if there is an y e A(F, E) such that y X = IE

( x y = IF).

x is called invertible if it is both left and right invertible.

284

1. Banach Spaces

The invertible morphisms of /: are precisely the isomorphisms of Banach spaces. Proposition

1.5.1.6

(~) ) Let A be a unital B a n a c h category, E , F

of A , and x an invertible m o r p h i s m of A ( E , F ) . y E A(F, E)

objects

Then there is a unique

with yx-

1E,

x y = 1F.

Put

X-1 : - - y . X -1 is called the i n v e r s e o f x .

Since x is invertible, there are y, z E A ( F , E ) so that y x -- 1E~

x z ~ 1F.

Then y = y l F = y ( x z ) = ( y x ) z = 1EZ = Z. Corollary

1.5.1.7

m

( 2 ) L~t (~,A) b~ a ~ t a t ,a~ach cat~go~y ~k~

E , F, G C ~ , and E _5+ F -~ G . I f x and y are invertible, then y x is invertible and (yx) -1 - x - l y - 1 .

We have (~-~y-')(yx)

- x-~(y-~)~

(y~)(x-~y -') - y(xx-')y

C o r o l l a r y 1.5.1.8

( 2 )

E , F c Y2, E ~ F , and F - ~

Let ( ~ , A ) E.

- ~-~x = 1~, -~ -

yy-~ -

1~.

m

be a unital B a n a c h category. Take

Then x and y are invertible iff x y and y x

are invertible.

By Corollary 1.5.1.7, if x and y are invertible, then x y and y x are also invertible. Assume now that x y and y x are invertible. Put U :-- xy~

V "-- y x .

Then ~ ( y ~ - i ) = ( ~ y ) ~ - i _ 1~,

so that x is invertible. Hence, y is also invertible.

m

1.5 Banach Categories

Definition 1.5.1.9

(

1)

(

2)

Let ([2, A ) b e a B a n a c h

(E,F),

285

system. The map

~ A(F, E)'

defined on E22 is a Banach system. It is called the dual of A and is denoted by A t. The dual of .At is called the bidual of .4 and is denoted by A" and the dual of M" is called the tridual of A and is denoted by A'".

Definition 1.5.1.10

( 1 )(2) (3) Let ~ be a c l a s s a n d A a Banach category (unital Banach category) over E2. A left A - m o d u l e (unital left A -

module) is a Banach system .4 over $2 together with a left multiplication over A such that (ab)x = a ( b z ) ]or every E, F, G, H c E2 and

E4 F4 G4 H A

A

A

(and such that the unit of A is the unit for the left multiplication). The right A - m o d u l e (unital right A - m o d u l e ) is defined in a similar way.

A is a left and a right A-module (unital left and unital right A-module) in a natural way. Proposition 1.5.1.11

( 1 )

(2)

Let (r

category (unital

Banach category), A a left A-module (unital left A-module), and H a Banach space. Given E, F, G c Y2 , put A H ( E , F) := s

ua : A ( G , E )

>H ,

E), H) ,

x,

>u[ax]

for all E--~ F 2+ G. Then the Banach system .A H with the above multiA

AH

plication is a right A-module (unital right A-module), and, similarly, if we interchange left and right. In particular, A ~ = A' is a right A-module (unital right A-module).

It is easy to see that the maps defined form a multiplication. Take E , F , G , D E I2

286

1. Banach Spaces

and E4

A

F~

A

G--% D. AH

Then (u(ba))[x] = u[(ba)x] = u[b(ax)] = (ub)[ax] = ((ub)a)[x]

for every D--~ E (and A

(ule)[y] = u[lvy] = u[y]

for every D --~ G ), so that A

u(ba) = (ub)a

(ule = u).

m

Definition 1.5.1.12

( 1 ) (~) ) ( ] ) Let ($2,A), (Y2,A)be two Banach categories (unital Banach categories). A (A, A)-module (unital (A, A ) module) is a Banach system over $2 endowed with the structure of a left A module (unital left A-module) and right A-module (unital right A-module) such that the left and the right multiplications are compatible. A A-module (unital A-module) is a (A, A)-module (unital (A, A)-module).

Every Banach category (unital Banach category) A is a A-module (unital A-module). Corollary 1.5.1.13

( 1 ) ( 2 ) Let (~, A), (~, A) be two Banach categories (unital Banach categories) and A a (A, A)-module (unital ( A , A ) module). Then fit' is a (A,A)-module (unital (A,A)-module). If A is a A module (unital A-module), then A' is also a A-module (unital A-module). By Proposition 1.5.1.11, ,4' is a left A-module (unital left A-module) and a right A-module (unital right A-module). Take E, F, G, H C ~Q and E4

A

F-~ G4 A' A

H.

Then (x, (bx')a>- (ax, bx'>- ((ax)b,x'>- (a(xb),x'> = (xb, x'a> = (x,b(x'a)>

for every x C A(H, E). Thus (bx')a = b(x'a),

m

1.5 Banach Categories

Definition 1.5.1.14

287

( 1 ) ( 2 ) ( 3 ) Let A be a B a n a c h category (unital

Banach category). A A-category (unital A - c a t e g o r y ) is a A-module (unital A-module) A

endowed with an inner multiplication (with a unit) such that

each left multiplication on A is compatible with each right multiplication on .4 and that the left and the right multiplication on .A over A are simultaneously compatible with the inner multiplication on yl.

A is a A-category (unital A-category). Every A-category (unital Acategory) is a Banach category (unital Banach category) with respect to the inner multiplication.

288

1. Banach Spaces

1.5.2 Functors Definition 1.5.2.1

(1)

(2)(3)

(~,A),(~,B)be two Banach

Let

systems. A f u n c t o r of M into B is a map f defined on f22 such that f(E,F) c s for every E , F

C E2. The functor f

is called isometric if f ( E , F )

is an

isometry for every E, F C f2. ,4 and B are called isometric if there is an isometric functor of A into B. The map (E, F ) ,

> 1A(E,F)

defined on ~22 is an isometric functor of ,4 into ,4. It is called the identity f u n c t o r of .A. Let ($2,A),

(E2,B), (f2, C) be Banach systems, f a functor of fit into B,

and g a functor of B into C. The map (E, F ) ,

~ g(E, F) o f ( E , F)

defined on ~22 is a functor of .4 into C. It is called the composition of the f u n c t o r s f and g and it is denoted by g o t . Let

(~,M) be a Banach system. Given E, F C ~ , let jEF denote the M(E, F ) . Then the map

evaluation on

(E, F) , ,', jEF defined on f2 2 is a functor of ,4 into its bidual A " . It is called the evaluation f u n c t o r of A . Given E E ~2, put jE Let (~2,.A), ($2, B)

:-- jEE.

be Banach categories (unital Banach categories). A

f u n c t o r of B a n a c h categories (unital B a n a c h categories) of A into B is a functor f of M into B such that f(xy) = f(x)f(y)

(and

f(1E)= 1E)

for every E, F, G E ~2 and

E-5 F 4 A

.4

C.

Let ($2, A) be a Banach category and ,4, B left (right) A-modules. A funct o t of left (right) A-modules of A into B is a functor f of A into B such that

1.5 Banach Categories

f (ax) = a f (x)

289

( f (xa) = f (x)a)

for every E, F, G c $2 and E4

A

F4

A

G.

(E--% F-5+ G ) . A

A

A f u n c t o r of A-modules is a functor of left and right A-modules. Let ($2, A) be a Banach category (unital Banach category) and let .4,13 be two A-categories (unital A-categories). A functor of (unital) A-categories of .4 into 13 is a functor of A-modules of .4 into 13 which is also a functor of Banch categories (unital Banach categories).

Example 1.5.2.2 Banach system

( 2 )

Let $2 be a class of Banach spaces and `4 the

(E,F),

>s

over $2. Given E, F C $2, put

/(E,F):L(E,F)

~A(E,r),

~,

~ ~'

Given E, F, G C ~2 and E--~ F 2+ G , put A

A

V?.t : z

?J, O V

.

Then `4 with this multiplication is a unital Banach category and f is a functor of unital Banach categories of E into `4. `4 is called the transpose unital category of s and f the transposition f u n c t o r of s

The result follows from Theorem 1.3.4.2 and Corollaries 1.3.4.3, 1.3.4.4, and 1.3.4.5. I Proposition 1.5.2.3

( 2 ) Let ($2, A) be a Banach category. Let `4, B be

left (right) A-modules and let f be a functor of left (right) A-modules of .4 into B. Given E , F C F2, define f ' ( E , F ) = f(F, E)'. Then f' is a functor of right (left) A-modules of B' into A ' , called the transpose of f .

290

1. Banach Spaces

Given E, F C .Q,

Take E, F, G E .Q and

E-~ F-~ c. A

B'

Then (x, f ' ( y ' a ) ) - ( f x , y'a) - ( a / x , y') - ( f ( a x ) , y') - (ax, f ' y ' ) = (x, (f'y')a)

for every x C .A(G, E) (Theorem 1.3.4.2 a)). Thus f'(y'a) = ( f ' y ' ) a .

Proposition 1.5.2.4

( 1 ) ( 2 )

I

Let ($2, A ) b e

a Banach category and

,4 a left (right) A-module. Then the evaluation functor on ..4 is a ]unctor of left (right) A-modules into its bidual.

Given E , F E ~2 and x c A ( E , F ) ,

set

x= j A x .

Take E, F, G c f2 and E - ~ F - ~ G. .,4

A

Then (6"-~, x') - (ax, x') = (x, x'a) -- ('~, x' a) = (a~, x')

for every x' C A'(G, E), so a'-~ -

Definition 1.5.2.5

a~.

I

( 1 ) ( 2 ) Let (f2, A ) b e a B a n a c h

a left (right) A-module. Take E , F , G c ~2. Given E-~ F 4 G A'

( E-5,. F S~ G),

A

A

A'

set x x " A(C, E) ---+ IK ,

(~'~ A(a,E)

)~,

a,

) (ax, x') ,

, )(xa, x'>).

category and .4

1.5 Banach Categories

291

If A = A, then it is easy to see that the above composition law coincides with the multiplication introduced in Proposition 1.5.1.11. P r o p o s i t i o n 1.5.2.6 ( 1 ) ( 2 ) Let A be a B a n a c h category a n d " 4 a left (right) A-module. The composition law introduced in Definition 1.5.2.5 is an (A', A, A')-multiplication ((A, .4', A')-multiplication) such that X !

E --+ F _5+ G -% H ~ ( resp. E --% F _2+ G -~ H ~ E4

x !

F-+

(resp. E ~

F-~

G4

( a x ) x ' = a(xx') (x' x)a = x' (xa) ),

H==~ ( x x ' ) a = x(x'a),

G--~ H ~

(ax')x=a(x'x)

),

where a is a morphism of A, x is a morphism of A, and x' is a morphism of "4'. If A is a A-module, then X !

E _5+ F --~ G -+ H ~

(x'a)x = x'(ax),

E -~ r - - ~

(xa)x' = x(ax') ,

G ~

H ~

with the same conventions as above. The first assertion is easy to verify. We have

(b, (ax)x'} = (b(ax),x'} = ((ba)x,x'} = (ba, xx'} = (b,a(xx')} (resp. (b, (x'x)a} = (ab, x'x} - (x(ab),x'} = ( ( x a ) b , x ' ) = (b,x'(xa)}),

(b, (xx')a} = (ab, x x ' } -

((ab)x,x'} = (a(bx),x'} = (bx, x'a} = (b,x(x'a)}

(resp. (b, (ax')x) = (xb, ax') = ((xb)a, x'} =

= (x(ba), x'} = (ba, x'x} = (b, a(x'x)} ) for every H - ~ E , which proves the relations. If A is a A-module, then A


II

292

1. Banach Spaces

Corollary 1.5.2.7 ( 1 ) ( 2 ) I r A is aBanachcategoryand.4 (right) A-module, then

(jFax)x' = xx'

isaleft

( x t ( j E F z ) = XtX)

for all objects E, F, G of A and E s

F-f+ a

A'

(E-Y+ F S~ G) .

A

A

A'

Given a C A(G, E)

(a, (jFax)x') = (x'a, jvax) = (x, x'a) = (ax, x') = (a, xx') ((a, x'(jEFX)) = (ax', jEFX) -- (X, ax') -- (xa, x') = (a, x ' x ) ) . Thus (jvG~)~' = ~ '

Definition 1.5.2.8 ( 1 ) ( 2 )

(z' ( j . p x ) = z'z).

I

Let (Y2, A) be a Banach category and .4

a left (right) A-module. Define >IK,

x'a" : A(G, E)

x,

> (a",xx')

z,

~ ~",~'x>

for E, F, G C .Q and

z~4 F4a A" A' a"x' A ( G, E ) ----+ IK, J

for E, F, G E f2 and

E4A' F~A" a). It is easy to see that if .4 = A, then the above definition coincides with Definition 1.5.2.5. P r o p o s i t i o n 1.5.2.9 ( 1 ) ( 2 ) Let A be aBanach category, .4 a left (right) A-module, and E, F, G, H objects of A . a)

The composition law introduced above is a right (left) multiplication on .4' over A" with the property that E ~-~ F - ~ G _5+ H ~ A"

A

A'

A'

A

A"

1t

x(x'a")= (xx')a"

1.5 Banach Categories

b)

293

If F - ~ G and A'

u "A(G,H)~

A'(F,H),

(u " A ( E , F )

~ A'(E, G) ,

X l

x,

} XX t

". x'x) ,

then

H~ F~u~a "=x'a" Au

(G~ E~u'a"=a"x'). All a) The first assertion is easy to verify. By Proposition 1.5.2.6, (a,x(x'a")l - ( a x , x'a") -- (a", (ax)x' I - (a",a(xx')) = (a, (xx')a")

((a, a" (x'~)) = (a", (~'z)a) - (a", ~' (~a)) -- (xa, a"~') - (a, (a"~')~)') for every H-% E . A

b) We have (x, ulall I ~_ (~tx~a It) - - ( x x l

a 1'} - - ( x ,

xPall I

for every G _5+H A

((x, ~'a") = ( ~ , a") -- (~'~, a") - (~, a"x') for every E-G F ) .

W

A

(1) (2)

Let (~,A)

b e a B a n a c h cat~go~y. 1)e~ne

x"~"-A'(G,E)

)~,

~',

)(y",~'x"),

z" ~ y"- A'(G, E)

~~ ,

~',

~ (~", y"~')

Definition 1.5.2.10

for E , F , G c ~ , EV-~ F ~-~ G. A"

An

-q and ~ are called the right and the left

A r e n s multiplication, respectively (1951). Remark.

q and F- may be different (see Remark to Example 1.5.2.15).

294

1. Banach Spaces

(i)(2)

Proposition 1.5.2.11 Let A be a B a n a c h category, E , F , G , H objects of A, and A a left (right) A-module. Then a pt

E--+ F -~ G Y+ H ::=::vx ' ( b" -q a " ) = (x' b" ) a" A"

(E --+ F .A t

.A'

A pt

-~ G ~-~ H ~ A"

(b" ~- a " ) x ' = b"(a"x')

A"

).

By Proposition 1.5.2.9 a) <x,x'(b" q a")) = (b" q a",xx') = (a", (xx')b") = ( a " , x ( x ' b " ) ) - {x, (x'b")a")

((x, (b" F- a")x') = (b" F- a",x'x) - (b",a"(x'x)) - (b", (a"x')x) = (x,b"(a"x')))\ for every H -5+E.

m

A

Theorem 1.5.2.12

( 1 ) (2)

Let A be aBanachcategory.

a) q and ~ are inner multiplications on A". b) A" together with q (with ~-) is a A-category. We denote this category by A'~ (A~). The evaluation functor of A into A'~ (into A'~ ) is a functor of A categories.

d) If A is a left (right) A-module, then ,4' is a right A'~-module (left A~module). e) A' is a left A'~-module and a right A~-module.

f) If A' is a (A~,A~)-module, then (z" q y") ~ x" -- z" -~ (y" ~ x") for all objects E, F, G, H of A and E~F~GS~H. Apt

Apt

Au

g) If A is a unital Banach category, then A'~, A'~ are unital A-categories. In this case, the evaluation functor of A into these unital categories is a functor of unital A-categories, and A' is a unital left A~-module and a unital right A~-module.

1.5 Banach Categories

h)

295

If j denotes the evaluation functor of A, then x"x -- x" ~ (jEFX) -- x" ~ (jEFX) ,

y x " - - (jGHY) ~- x" = (jGHY) ~ X" for all objects E, F, G, H of A and x I!

E-~ F-+ A

i)

Let E , F , G , H

G -~ H .

A"

A

x/!

be objects of A and take F ~

G. Then the maps

Atl

A"(E,F)

>A " ( E , G ) ,

y",

>x" q y " ,

A"(G,H)

>A " ( E , G ) ,

y",

>y"~-x"

are continuous with respect to the topologies of pointwise convergence. It is easy to see that -~ and ~ are (A", A", A")-multiplications. Let E, F, G, H be objects of A. In order to simplify notation, we adopt the convention that the symbols x, y denote morphisms of A, the symbol x' denotes a morphism of A', and the symbols x", y", z" denote morphisms of A". a) By Corollary 1.5.2.7, A" is a A- module. Let

-+H. By Proposition 1.5.2.11, ((z" ~ y") ~ ~ " , x ' ) = ( x " , ~ ' ( z " ~ y")) - (x", (x'~")~"l -

= (y" ~ ~", ~ ' z " ) - (z" ~ (y" ~ x"), ~'), ((~" ~ y") ~ ~",~') = (z" ~ y",x"~') = (z",y"(x"x')) = (~", (y" ~ ~")x') = (z"~ (y"~ ~"), ~'), x !

for every H - + E . Thus

(z" -~ y") ~ x" -- z" -t (y" ~ x") ,

296

1. Banach Spaces

(z" ~ y") ~ x" -- z" ~ (y" ~- x").

b) Step 1

E -~ F ~ F-5+H ~ "

x(y" ~ x") - (xy") t- x" i x ( y " ~ x") - (~y") ~ ~"

By Proposition 1.5.2.9, (x(y" ~ x"), x') - (y" -~ ~", x'x) - (x", (x' z)~"l -

= (x", ~'(xy")) - ((zy") ~ x", x ' ) ,

(~(y" ~ ~"),~') - (y" ~- x " , x ' z ) = ( ~ " , x " ( x ' x ) ) -

= (y", (x"x')~) = ( x y " , x " ~ ' ) =

( ( x y " ) ~ x",~')

for every H -~ E.

st~p 2 E-~F -~ r - ~ H ~

{ ( ~ " -~ x")~ - y" -~ (x"x) (y"~ x")x - ~"~ (~"~:)

Given H - ~ E ,

((y" ~ x")x, x') - (y" ~ x", xx') = (z", (~x')y") = (:~", :~(x'y"))= (x"x, x'~")= (y"-~ (x"~), x'), ((y" ~ x")x, x') - (y" ~- ~", ~ ' )

= ( y " , ~ " ( : ~ z ' ) ) = (y", ( x " x ) x ' ) -

-

(y"~ (x"~),x')

(Proposition 1.5.2.6). Step 3

E~

-~ G ~ H ~

~ (Sx) (Y"x) ~~ ~x "" -- y"~ y'' ~ (xx") (~") (

Given H - ~ E , ((y"~) ~ ~",~') - ( ~ " , ~ ' ( y " ~ ) ) = (~", (~'y")z) - ( ~ " , z ' y " )

- (y" ~ ( ~ x " ) , ~ ' ) ,

((y"~) ~- ~",~') - (y"~,~"~:') - (y",~(~"~')) - (y", (~x")~') = (y" ~- ( x ~ " ) , x ' )

1.5 Banach Categories

297

(Proposition 1.5.2.6). Step 4 b) Follows from Corollary 1.5.1.13 and the preceding steps. c) By Proposition 1.5.2.4, the evaluation functor j of A is a functor of A-modules. Take E ~ F--~G. Then (j(yx),x')-

(y(jx),x')=

(y, ( j x ) x ' ) -

(jy, (jx)x'} = ( ( j y ) ~ ( j x ) , x ' l ,

( j ( y x ) , x ' I - ( ( j y ) x , x ' } - ( x , x ' ( j y ) } - (jx, x'(jy)) - ((jy) ~ ( j x ) , x ' ) , for every G - ~ E , so that j ( y x ) = (jy) ~ (jx) - (jy) -~ ( j x ) . d) follows from b) and Proposition 1.5.2.11. e) follows from Corollary 1.5.1.13 and d). f) Given H - ~ E ,

((z" ~ ~") ~ ~",~') = (z" ~ y",x"x') - (y", ( ~ " ~ ' ) ~ " ) = = (y",~"(~'z")) = (y"~ ~",~'z") - (z" ~ (y"~ x"),~'). x II

g) Take E ~ F .

Then

( x " ~ (jE1E),X') - (X", (jE1E)X'} - (X", 1 E X ' ) = (X",X'},

((j~l~)

~ ~",~'1 -

( j . l ~ ,x"x') = (1 ~, x"x') = ( X " , x ' l g ) -

(x",X'),

(x" -~ (jE1E), x') = (jE1E , X'X") -- (1 E , X'X") = (X" , 1EX') = (X", X'} , ((jrlF)-t x",x')=

(X",x'(jF1F))-

(X",X'IF) - (X",X')

for every F---~E (Corollary 1.5.2.7, Corollary 1.5.1.13). Thus X" ~- (jE1E) -- X",

(jF 1F) ~ X" = X",

x" -t (jE1E) = x",

(jF1F) -~ X " = X".

Hence A"~ and A"~ are unital A-categories and the evaluation functor of A into these categories is a functor of unital A-categories.

298

1. Banach Spaces

h) We have

(x" F- (jEFX), X') = (X", (jEFX)X') = (X", XX') = (X"X,X') , (x" -d (jEFX),X') -- (jEFX, X'X") -- (X,X'X") = (X"X,X') for every G - ~ E and

( (jaHY) F- X", X') = (jagY, X"X') -- (y, X" X') -- (yx", X'), ((jaHY)-d X",X')= (X",x'(jGHy))= (X",X'y)= (yx",x') ggt

for every H--+F (Corollary 1.5.2.7). i)We have lira (x" --t z",x') -

Ztt---~ytt

for every E ~ F ,

lim (z",x'x") - (y",x'x") = (x" --t y",x')

z"---+ytt

G -~ E , and

lim (z" ~ x", x') = ztt--+ytt lim (z", x"x') = (y", x"x') = (y" F- x", x')

ztt--._~ytt

tt

for every G ~ H ,

Xt

H -+ F .

m

P r o p o s i t i o n 1.5.2.13 Let A be a Banach category and E, F , G , H objects of A. Take F Art ~-~G. Let j denote the evaluation functor of A' and put

u=, : A(F, G)

~, A'(E, G) ,

(resp. u:, "A(F,G)

x,

~. A ' ( F , H ) ,

~ xx' x,

~ x'x)

for every E -~ F (resp. G -~ H ). Then the following are equivalent: At

a)

At

x It

E --+ F =~ x" ~ y" = x" -~ y" AH

(resp. G Art ~-~ g =~ y" F- x" = y" -t x" ). b)

The map A"(E,F) ( resp.A" ( G, H)

~ A"(E, G), >A" ( F, G) ,

y", y" ,

; x" ~ y" > y" -t x")

is continuous with respect to the topologies of pointwise convergence.

1.5 Banach Categories

c)

299

E s p ~ ~".x" = j~(x"~') At

(~p.

II x l l c ~3s H ~ u~, = j~(~'~")

).

A~

d)

E -~ F ~ ux,"x" e jEc(A'(E, G)) At

(~p. a ~AH t

II II ~ ~,~ 9 j~,(A'(P,H))).

a ~ b follows from Theorem 1.5.2.12 i). b ~ a. Let i be the evaluation functor on A. Then iEF(A(E, F)) (resp. icH(A(G, H ) ) ) is dense in A"(E, F) (resp. in A"(G, H ) ) with respect to the topology of pointwise convergence (Corollary 1.3.6.5). By Theorem 1.5.2.12 h), x" F- y" - x" A y"

(resp. y" ~ x" = y" -~ x")

for every y" e iEF(A(E,F)) (resp. y" e iaH(A(G,H)) ). By continuity, ( b ) and Theorem 1.5.2.12 i)), x" k y" = x" q y"

(resp. y" F- x" = y" -~ x")

for every E - ~ F (resp. G - ~ H). AH

A"

a :=~ c. By Theorem 1.5.2.12 h) and Proposition 1.5.2.9 b), (y", ~," ~") = (~", ~x,y' "'~ = (x", ~'y") = (y" ~ x", ~') =

= ( y " ~ x",x') - (y",x"x'} -- (y",jEG(x"x'))

for every G -~ E (resp. AH

(~". ~".~") - (x", ~'~,y) - (x", y"~') - (x" ~ y", x') = - (~"~ y " . ~ ' ) - (y". ~'~")= (y". j~.(x'x")) p!

for every H ~,, F ), so tl X t t

u x,

- jEa(X"X')

It tl (resp. ux, x -- jFu(X'X")).

C =~ d is trivial. d ==> b. By Proposition 1.5.2.9 b), ( ~ " ~ y". ~'} - (x". ~" ~'} - (x" . ~.' y "'~ = ~ . '" ~". ~")

for every E -~ F and G -~ E A"

AI

300

1. Banach Spaces

(resp. (y" -t x", x') - (x", x'y") - (x", u'x,, y"> -
A'

A"(E,F)

>A " ( E , G ) ,

(resp. A"(G, H)

y",

> A"(F,G),

> x " ~ y" y",

> y" -~ x")

is continuous with respect to the topologies of pointwise convergence,

m

P r o p o s i t i o n 1.5.2.14 Let (~2, A), (~2, B) be two Banach categories and let f be a functor of Banach categories of A into B. a)

I I E _5+ F 13' v_~G ::v (fFaY )x A

~

I fsa(Y'fEFX)

l I E s13' F 4,4 G :=~ x(fEFY ) = f ~I a ( ( f F a x ) y I ).

b)

1 I E ~ F ~ G ~ (fFay )x" A"

__

13'

till'!

I

E 13' v_~F A-~" G ~ x tJEFY ) EG ~ ~ a ~ .4" .A"

__

J"' E G t Y" l

l/tit

'"" JE, F x "") , It

IEG ((JFa x )Y').

f ~u ( Y " ~ ~'') - ( eJu ~ Y u ~J ~ "( ]~ E" F x " ) f~c(Y" ~ x") - (]i:aY"" ") -~ t.JEF "),

i.e. f" is a functor of Banach categories of .A~ into 13~ (of Jt~ into r2,,~ a) Take G --%E. Then A

(a, (f'ray')x) - (xa, f ' p a Y ' ) - (far(xa),y') - ((fErx)(fc, E a ) , y ' ) = (/~a,

y ' f ~ . ~ > - (a, f ~ ( y ' / ~ ) >

in the first case and (a,x(f~y'))-

(ax, f'EPY')- (fpE(ax), y ' ) -

((faEa)(fpax),y')-

= (faEa, (frax)y'} -- (a, f'~a((fpax)Y')) in the second case. b) Let G - ~ E . By a), A

(X, I''

(JraY ,~)x ,, )

_

( X u,

xfFay') = (x", I'FE((faEx)Y')}

=

1.5 Banach Categories

__

-

301

I1 :EFx , (faEx)y') = (faEX, y ' :"E' : " ) = (x, :'Ea' ~y"'"':Erx"')>

( I'll

in the first case and (x , x" f 'EF y'X, = (x" , (fEFY)X) ' ' ' ' faEX)) = (x" , fGF(Y _- P:lFl a x It .y I faEx) -- (faEX, (:" :..: "")y')

-

-

I /I'll It (x.:'~.(~:,~: )y')>

in the second case. X! c) Let G---+E. By b), A'

(f~.(y" ~ ~") .~') = (y" ~ ~,,. : ~,: ,> - (~/.,~"":~:~'" =

--

Y", ."IF fG

II

II

!

((fEFX )X ) } -

(~11 / I I JEG(Y __-- ( X l l ~ J F E ~ /

-I

Xlt

I :FGYPII

E x a m p l e 1.5.2.15

~i'll

II

11

/!

1

//tl!

!1\

~,:FaY , (f~Fx )X ) - - ~,(JFaY ) ~ (f~Fx"), x'),

), x') =

(y#

I!~)> __ ~ JrEl l F

I!

p

I

~ x , f 3 ~ x ) -- (x", (f~E ')y") = !1 ~ xlPI!jFGylI~)

= ~(]FGYl/rll

II) ~1 ~ ] E F

I1~)~ x l >

.

B

Let ~2 be a class of sets and A the Banach system

(s, T) . ~ :,~(S, T) defined on $22 endowed with the multiplication defined in Proposition 1.2.3.5. Take P, R, S, T e ~2 .

a)

A is a Banach category. For convenience, we settle on the following notations: 1) h , k

denote morphisms of A ,

k' denotes a morphism of

A' and h", k", 6" denote morphisms of A" ; 2) p, r, s, t denote points of P, R, S, T , respectively.

b)

A' may be identified with the A-module

(s, T) ~. > ~',~176 T) , kk" T x R

~, IK,

(t, ~) ,

~~

k(~, t)k'(~, ~) 8

for R - ~ S -~ T , and k'k" T x R - - + lK,

(t, ~) ~

~

k(~, ~)k'(t, ~) 8

fo~ R - S S ~ T .

302

1. Banach Spaces

c) A" may be identified with the map ( S ,T) ,

>E(S, T) ,

where E ( S , T ) is the Banach space introduced in Proposition 1.2.3.7 a) for p - 1 and

cl)

R ~ S -~ T :=v (kk")r

:

f

k(s,-)kT({s}) ,

k(s, .)dkT(s ) - E 8

k II

R -~ S -~ T ~ (k"k)r- ~

~)

k(~, ~)k;', 8

c3) R -+ S -~ T =~ ( [t, r] = k'(t, )dk~r' ~" k'k") f 9 , R -~ S ~-~ T ~

C4)

(kt'kt)[t, r] -- E

k'(s,r)k;'({t}), 8

(h" q k")r - fhTdk'~'(s)

R~ s ~ T~

%)

(h" ~- k")~- E k"({~})h~. 8

d)

A~ is a unital Banach category. For t C T, and A' is a unital left A~-module.

e)

A' is a

(1T)t

iS Dirac measure at t

(AT,A~)-module.

f) P 5 R g S ~4 T ~ (h" ~ k") ~ e" = h" ~ (k" ~ ~")

a) is easy to see. b) The identification follows from Proposition 1.2.3.6 c). Given T --~ R, (h, kk') - (hk, k') - ~ ( h k ) [ ~ ,

~]k'(~, r) -

8~r

= E (Ek(s't)h(t'r))k'(s'r) s,r

- Eh(t,r)(Ek(s,t)k'(s,r)),

t

t,r

/h, k'k) = (kh, k') = ~ ( k h ) [ t ,

s

s]k'(t, ~) -

t~s

E ( E h(t,r)k(r,s))k'(t,~)- ~ h(,, r)(~ k(~,s)k'(~,~)) t~8

r

t,r

s

1.5 Banach Categories

303

c) The identification follows from Proposition 1.2.3.7 b). We have:

c,)

T -+ kR~

: :~ r

i

@k", k')= @", k'k) = ~

S(E ~(',')~'(','))"~"(')-

(k'k)(~, ~)dk"(~)=

r

t

-~ ~,(..,)(I~(,.,),~.,(,))r~t

r,t

r

s

T -+ R ~

(k"k, k')= (k"kk')= ~

(kk')[~,t]dk'~'(t) = 8

8

r

r

8

c3) r 4 e ~

(k, k'k")= (k", kk')= Z / ( k k ' ) [ ~ , ~]ek'r'(~)= 8

-E

s

S(~ ~("r)~'("'))'~"(')t

t,r

c4) T & R ~

(k, k"k') = (k", k'k) = ~ f(k'k)[s, t]dk2'(t) = 8

- ~ S(v- ~(,..)~'(,..)),~'.' (~)- ~ ~,(...)(S~(,..),~,.,(,))__ 8

r

8~r

8~r

t

t,r

s

c5) r -~ ~' R ~

(h" ~ k" , k')= @", k'h") = ~ S @'h")[~,~]ek"(~) = r

r

r

= (h" V k", k') = (h", k"k') = E J(k"k')[s, t]dhT(t ) = 8

8

r

: Efk'(~, )d(E k'r'((~)h:) r

8

304

1. Banach Spaces

d) We have (1r -t k")s -- f f (1T)tdk:'(t) = k~',

(h" ~ lr)~ = f h',:d(lr)~[t'] - h~ k"

h"

for S --+ T --+ R and S -~ T. e)

P -~ R -~ S -~ T ~ : f(E

((k"k')h")[t,p]-

k'(s, r ) k : ( { t } ) ) d h ; ( r ) 8

= E

E

f (k"k')[t,~]dh';(~) (Jk'(s, r)dh;(r))k:({t})-

$

(k'h")[s,p]k;'({t})- (k"(k'h"))[t,p]. 8

f) follows from e) and Theorem 1.5.2.12 f). Remark.

If k T ( { s } ) - 0 for every (r,s) e R x

S incs),then

h" t- k" = 0 for every h". Take S -~ S and for s E S , let hi~' be the Dirac measure at s.

Then h" -t k" -- k" by d). Hence t- and --t may be different. Definition 1.5.2.16 ( 2 ) Let (X?,A) be a Banach category and .4 a A module (A-category). A A-submodule (,4 subcategory) of .4 is a A-module (A-category) 13 such that B(E, F) is a Banach subspace of .4(E, F) for every E, F C ,(2 and such that the multiplications of B are the restrictions of the multiplications of .4. The map defined on f2 2 such that f (E, F) is the inclusion map

u(E, F)

~A(E, F)

for every E, F C ~2 is called the inclusion functor of 13 into .4. P r o p o s i t i o n 1.5.2.17 ( 2 ) Let (O,A) be a Banach category. Let ,,4 be a A-module (A-category) and 13 a A-submodule (A-subcategory) of A . Put

C(E,F)

:=

A(E,F)/U(E,F)

1.5 Banach Categories

305

and let f (E, F) denote the quotient map ,,4(E, F)

. ~ C(E, F)

for E, F E ~ . Define multiplications on C by taking the factorizations. Then C is a A-module (A-category) and f is a functor of A-modules (A-categories). C is called the quotient A-module (quotient A - c a t e g o r y ) o f gl by 13 and is denoted by A/13; f is called the quotient functov of A onto A / B . Let (O,A) be a unital Banach category. If A is a unital A-module, then A/13 is a unital A-module. If A is a unital A-category, then A/13 is a unital A-category and the quotient functor of A

onto A/13 is a functor of unital

A-categories. The proof is a long verification. P r o p o s i t i o n 1.5.2.18

( 2 )

I

Let (a2, A) be a Banach category, .4 a A -

module, and 13 a A-submodule of .4. Given E, F E $2, set B~

:= 13(F,E) ~ .

Then 13~ is a A-submodule of .A' . For E, F E g2 , let f (E, F) denote the canonical isometry (.A'/B~

>B ' ( E , F )

(Proposition 1.3.5.2). Then f is an isometric functor of A-modules of A'/13 ~ into 13'. Take E, F, G, H E ;2 and consider x !

E-~F--+G--~H. A

B~

A

Then {m,m'a) =

(az, z'}

= O,

(y. b.') = ( v b . . ' ) for every

G-~E, 13

so that

H-~F 13

= 0

306

1. Banach Spaces

x'a C B ~

bx' C B ~

Hence B ~ is a A-submodule of A'. Take E, F, G, H E $2, X!

E-~F--+G-2+H, A

.,4'

A

and let q be the quotient functor of ,4' onto .A'/13 ~ . We have

(x, f ((qx')a)) = (x, (qx')a) = (ax, qx') = (ax, f (qx')) = (x, (f (qx'))a) , (y, f (b(qx'))) - (y, b(qx')) = (yb, qx') - (yb, f (qx')) = (y, bf (qx')) for every

G-~E, B

H--~F. 15

m

This proves the last assertion.

Proposition 1.5.2.19 tion functor of .4, and

( 2 ) Let (~,.A) be a Banach system, i the evalua-

(~(A)~

r ) . - (~(.4)(r, E)) o

If M' is a Banach category and i(A) is a submodule of M", then i(A) ~ is an ,t ,,, _submodule. A~'-submodule and an .-'4 We denote by j the evaluation functor of .,4'. Stepl

E ~ F ~-~ G ~ H = = , x ' " x " = O , y " x " ' = O .

We have

(~', x'" x") = (~'", x"x') = o,

(y', y"~'") = (~'", y'y") = o

for every

G-~E, A'

Step 2 E ~

F ~ G ~ i(.4)o ~(.4)o

For G - ~ E ,

H Y-~F. A'

y'" F- x'", y"' -~ x'" c (i(A)~

G) .

1.5 Banach Categories

307

(y'" ~ x'", x") - (y"', x'" x") = O,

(y"' -~ x'", x") = (x'", x"y"') - 0

by Step 1. Step3 E ~ j(A')

F Y-~ G ~-~ H===v i(A)

j(A')

~

y,! ~ x , l y,! -d x "l, z "l ~- y , t z m --t ym are morphisms of

i(A) ~

Take x !

zI

E~F,

G-+H

A'

A'

with XlI!

--

9 JEFX

!

~

Zl!!

~

9 JGH

z

!

.

Then (y'" -~ x'", x") -- (x'", x"y'") -- O,

(y"' ~- x'", x") - (y"', x'" x") - (y"', ( j E r x ' ) x " )

(z'" ~ y"',y") -

-- (y"', x ' x " ) -- O,

(z'",y"'y") - o ,

(z'" -~ y'", y") -- (y'", y"z'"} -- (y"', y " ( j a H Z ' ) ) = (Y'", y"z') -- 0

for every

G~E, HY--~F i(A) i(A) by Step 1 and Corollary 1.5.2.7. Step 4

i ( A ) ~ is an A~'-submodule and an A~'-submodule.

The assertion follows from Steps 2 and 3 together with Proposition 1.3.6.19 d). m

308

1.6

1. Banach Spaces

Nuclear

Maps

Several classes of compact operators on Hilbert spaces are known. These classes have some connection or other to the eP-spaces (p E {0} U [1, oc[). The class of nuclear operators, which are also called trace operators, arises when p = 1. They are the subject of this section. Their strong properties make the theory also applicable to Banach spaces, as shown by Alexander Grothendieck. Since this theory is not pursued further in this book, the reader may skip this section. 1.6.1 G e n e r a l R e s u l t s D e f i n i t i o n 1.6.1.1

( 0 ) (Grothendieck, 1952) Let E , F

be normed spaces.

A map u" E -+ F is called nuclear if there is a family ((x'e, Y~))eE, in E' x F such that

eel

and

eEI

for every x E E . We write s

F) for the set of nuclear maps of E into F

and define

I1~ I ~ - i~f E I1~:11Ily~ll eEI

for u E E , I ( E , F ) , where the infimum is taken over all families ((x:,Ye))eE, in E' • F with the above properties. We set E,'(E) := f ~ I ( E , F ) .

We can replace the indexing set I by IN in the above definition, since

{~ E I I I1~:1111Y~II# 0} is countable. E x a m p l e 1.6.1.2 ( 3 ) L~t T b~ ~ ~t, E .-~0(T) F :-- ~P(T), p E [1, oo], y E F , and take u:E

~F,

x~

~xy.

Then u E f~l(E) iff y E gl(T) and in this case

Ilulll--Ilylll.

(~p. E . - ~ ( T ) ) ,

1.6 Nuclear Maps

309

Assume t h a t y c g l ( T ) . Given t c T , set

x t''E

)IK,

x,

)x(t)y(t).

Then

tCT

tET

E (x' x't}et - E tcT

x(t)y(t)et - ux

tET

for every x C E . Hence u is nuclear and

Assume t h a t u is nuclear. There is a family ((x'~, y~))~e, in E ' x F , such that

i~',ll i y, II < oo LCI

and

for every x E E . Then

LEI

so t h a t

LEI

~EI

for every t E T . Hence

Z ly(t)l ___~

~

tET

~CI

- ~ ~EI

tCT

i~, i~

I(~,, ~:)1 lly, i, =

I(~,, ~:)i _< Z Ily, II IIz:li,

tET

tEI

so t h a t y E gl(T) and

liyill _< ii~Jli 9

I

310

1. Banach Spaces

T h e o r e m 1.6.1.3

s

a)

( 0 ) Let E , F

be normed spaces.

is a vector subspace of f ~ / ( E , F ) and

for every u 9 s (E, F ) . The map

b)

C~(E,F)

>~+,

u,

>lull,

is a norm. Take f_.l(E, F) with this norm. c)

f_,I(E,F) is a dense set o f / ~ I ( E , F ) .

d)

/21(E,F) is complete whenever F is complete.

e)

u 9s

IIr

~ u' 9 s

llullx.

~

Let ((x't, yt))t e, be a family in E' • F

a), b), and c). Take u 9 s with

Ifx:fl Ily~fl < tEI

and such that

LGI

for every x E E . u is linear and

II<x,x:>y~ll = ~ I<x,~:>l Ily~ll ~<

Iluxfl ~< ~ tel

~< ~

tel

Ilxll IIx:ll Ily~ll - Ilxll ~

tel

I x:ll ly~ll

tel

for every x E E . Hence u is continuous and

JJ~JJ< II~fJ~. Given u,v c s

and c~ e IK, there are families ((x't,x~))~e, ,

((Y~,Ya))acL in E ' x F such that

tCI

AEL

1.6 Nuclear Maps

311

and 1iX ---

X~ X e I X

:

AEL

LE1

for every x E E . T h e n

~E I

~EI

,XE L

for every x C E . Hence u + v , c~u c E l ( E , F ) a n d

IlU -Jr-VII1 < IlulI1 -t-]t•111,

110~72111--10~ I I1~111 9

From

fi~rl~ - 0 , it follows t h a t

fl~lf = o a n d so

?~0. Hence

s

is a vector s u b s p a c e of s

ff.,l(g, F) ~

IR+,

and the map

U'

~ IlUI]I

is a norm. Given u C

s

a n d c > 0, t h e r e is a family ((z'~,y~))~ci in E ' x F

such t h a t

and =

LEI

for every x C E . T h e r e is a finite subset J of I with

~CI\J

Put

312

1. Banach Spaces

w'E

~F,

E

x,

~

(x,x, '}YL .

LEJ

Then

eCI\J

for every x 9 E . It follows t h a t u -

w is nuclear and

tEI\J

Since w 9 s

F), u 9 s

F) and s

F) is a dense set of s

d) Let (un)~e~ be a Cauchy sequence in s

F ) . By a) and b), (u~)nc~

is a Cauchy sequence in /2(E, F ) . P u t

u " - lim us n---~OC (Theorem 1.2.1.9 b)). We may assume t h a t

1 ii~n - u ~ + l I1 <

2,~

for every n 9 IN. Given n 9 IN, there is a family ((X'L, yn~))~Cin in E ' x F such t h a t

1 2n

tCI,~ and

(U n --Un+I)X--- ~ { X , XtnL)ynL rEin

for every x 9 E . Hence,

p-1 q=n tE Iq

for every n , p C IN, n < p , and x C E . Then , q=n ~E lq

and

0r q=n

1

1

1.6 Nuclear Maps

313

oo !

(un - u)x - unx - ux = lim (unx - upx) = E p---+oo

E (x' Xq~)yq~

q=n

I,E Iq

for n E IN and x C E . Hence u n - u C/21(E, F ) and 1

I1~-

~111 <

2n-1

for every n C IN. By a) and b), u is nuclear and (un)ne~ converges to u in

s

Hence s

is complete.

e) Let ((x'~,y~))~ei be a family in E ' x F such that

tEI

and

ux.-E(x,

x~,}yt,

t,E I

for every x C E . Then

tCI

tCI

for every (x, y') C E x F ' (Theorem 1.3.4.2 a), Proposition 1.2.1.16, Corollary 1.2.1.10, Corollary 1.1.6.10). Thus .

.

~'y' - ~ ( y ~ ,

r

t,E I

for every y' E F ' and the assertion now follows.

Coronary

1.6.1.4

I

( 0 ) rf E,F a,~ ,~o,~m~d ~pa~, th~,~ (., ~')y e e l ( E , F ) , I1(, x')yll~ = I1(, ~')yll = I1~'11 Ilvll

for every (x', y) C E' x F . Now

II(-,x')yll = sup II(~,x')yll = Ilyll sup I(x,~')l = I1~1111x'll. xEE#

xEE#

By Theorem 1.6.1.3 a), II(',x')Ylll-> IlYll IIx'll, and the reverse inequality is trivial.

II

314

1. Banach Spaces

Proposition u 6 s

1.6.1.5

( 0 ) Let E , F , G , H

9s

be normed spaces. Take

and w 6 s

Then w o v o u

9s

and

i ~ o v o ~il~ < li~il ilvillil~il Let ((y:, z~))~E, be a family in F' x G such that

Z

lly:ll Ilz~ll < oo

tEI

and

tel

for every y 6 F . Then

I1~% II Ilwz~ll <_ ~ tel

I1~,'11Ily/ll II~li II~li =

tel

= il~ll li~i ~

ily:ll liz~i < oo

eEI

(Theorem 1.3.4.2 b)) and

LEI

LEI

for every x 6 E (Proposition 1.2.1.16, Theorem 1.3.4.2 a.)). Hence w o v o u is nuclear and

li~ o v o ~ill __< II~li livil~li~li 9 C o r o l l a r y 1.6.1.6

m

Let ~? be a class of Banach spaces. The map on ~2 de-

fined by (E,F),

~s

together with the usual composition of maps as multiplication, is an s category. We denote it by f ~ ,

or simply, by f l

The assertion follows immediately from Theorem 1.6.1.3 a), b), d), and Proposition 1.6.1.5. C o r o l l a r y 1.6.1.7 p

m ( ] ) Let E , F be normed spaces, G a subspace of F ,

a projection of F onto G, and j the inclusion map G -+ F . Then

]]j o uli ~ < Ilul]~ _< ]IP]] fiN o ?_t]]1 for every u 6 1.1(E, G).

1.6 Nuclear Maps

315

We have u=pojou,

so that IIJ o ~11~ ~ Ilulll ~< Ilpll IIJ o ullx by Proposition 1.6.1.5. Corollary 1.6.1.8

i

( 3 ) Let E , F be normedspaces and take u E s

If F is a dual space, then

Ilulll = Ilu'lll

By Theorem 1.6.1.3 e), u' and u" are nuclear and I1~"11~ < ilu']il __ li~i]~.

By Proposition 1.3.6.19 b), there is a projection p of F" onto ImjF with IIPB]-< 1,so that

(Corollary 1.6.1.7). Since jF o u = u" o jE

(Proposition 1.3.6.16), Illtlll

=

IIJF o uIl~ = Ilu" o JEll1 ~ Ilu"ll~l[JFlI = I]~"ll~ ~ I1~'11~ ~ II~lll

(Proposition 1.6.1.5). Hence ll~ll~ = II~'ll~

Corollary 1.6.1.9

( 3 ) Let E , F be normed spaces and take u c s

If F is reflexive, then u' C s

and

Ilu'lll = Ilulll

Corollary 1.6.1.10

(

3

)

Let E , F

i

be normed spaces and take u C

f~l (E, F) . Then u' and u" are nuclear and

Ilu'lll = Ilu"lll

316

1. Banach Spaces

By Theorem 1.6.1.3 e), u' and u" are nuclear and by Corollary 1.6.1.8,

II~'ll~- It~"lll 9 P r o p o s i t i o n 1.6.1.11 Let E, F be normed spaces and G a subspace of E . Then, given u E s there is a v E E I ( E , F ) such that

~la=~,

I vii, = II~lll.

Let (( X !,, y,))bE, be a family in G' x F such that

IIx:ll Ily, II < tEI

and

LCl

for every x E G. By the Hahn-Banach Theorem, given t C I , there is a z~' C

E'

such that

z,'lG = ~,,'

I z,' I-

I1~,1 9

Set v'E tEI

Then v E E I ( E , F ) ,

vlG - ~,

II~ff~ _< Y~ IIz:l[ Ify, ff - Y~ If~',lf Ify, lf.

Hence

By Proposition 1.6.1.5,

~111 _< flvlli, so that

~11, : II~ll~ T h e o r e m 1.6.1.12

(3)

Let E , F

be normed spaces.

m

1.6 Nuclear Maps

a)

Given u E s

there is a unique ~ E s

317

such that

for every (x, y') C E x F ' .

b)

The map

is an isometry.

Step 1 a) and I 1 ~ 1 < 1 ~ Let ((x~, y~))<es be a finite family in E x F ' such that

y:)x~ = o.

Z(, We show t h a t

(~x~, x:) = o. Since the m a p E

~IK,

z~

>(uz, y')

is linear for every y' C F ' , we may assume that (XL)LCI is linearly independent. Given y c F ,

~(

' y, y~}x~ = 0

so that

(y, y:) - o for every y E F and c C I . Hence y~ - 0 for every c E I and so

(~x~,y : )

-

o.

By the above proof (and Corollary 1.3.3.4), the map

U-Cs(F,E)

>~ , ~ < - ,y<)' ~ <

> ~@x~

, y<) '

318

i. Banach Spaces

is well-defined. It is obviously linear. Now

I~(~<., ~:>~)I- I~<~x~,~:>l _~z ~<~x,,y:>~_~~ ~~x~ ~:~ ~EI

~EI

~

~EI

~EI

I1~11 Ix~ll Ily:ll --- lull ~ IIx~ll Ily:ll,

LEI

l,E I

so that

y~(., y:)x, ~,EI

tEI

for every finite family ((x,,y~)),e, in E x F ' . Hence u is continuous with respect to the norm I[" 111 on s E) and [lull < ]lull. By Theorem 1.6.1.3 c) and Proposition 1.2.1.13, there is a unique g E E,I(F, E ) ' , such that

~=u on /2s(F,E ) and

I1~1 = Ilull ~ I~11. Step 2

b)

Take 0 E/2 I(F, E)' and x E E . Set

~xF'

;~,

~',

>0((.,y')x).

Then J(ux, y')] = ]O((.,y')x)l ~ I]61l II(',y')x]ll = IlOll Ily'll Ilxll for every y' E F' (Corollary 1.6.1.4), so that ux E

F",

II~x[I ~ Ilell Ilxl.

Hence u E E,(E,F"),

IlulJ < IIOll.

By the definition of u, ~ - 0 on E,s(F , E) (Corollary 1.3.3.4). By Step 1 and Theorem 1.6.1.3 c), ~ = 0 and

II011- I1~11 < I1~11_< II01, I1~[I-lul.

I

1.6 Nuclear Maps

Corollary 1.6.1.13

319

Let $2 be a class of reflexive Banach spaces. Given

E, F C $2, let f (E, F) denote the isometry f_,(E, F)

> s (F, E)'

defined in Theorem 1.6.1.12 b). Then f s

is an isometric functor of the unital

L ~ into the dual of the unital s

s

Given E, F E $2 and u E/:(E, F), put U :--- f E F ? . t .

Take E, F, G e $2, u e / : ( E , F), and v e/:(F, G). Then

v~(~/.,

z:~) = E

~CI

,z:~ = ~~x~,v,z:l -

LEI

LCI

LEI

eCI

~(~, z:~)

tCI

- y~,z:~

: ~(u(E~,

- ~ ( E ~ , z:~)

-

~(E~,z:~x~)

z:~)) -

eEI

tEI

for every finite family ((x~, z[))~i in E x G' (Corollary 1.3.4.8). By Theorem 1.6.1.3 c), f (vu) : ~'~ - v~ - v f (u) ,

f (vu) - ~ - ~u - f (v)u ,

I

which proves the assertion. Proposition 1.6.1.14 s

( ] )

Let E , F

There is a unique ~ E s

be normed spaces and take u C such that

~( (., z')y) = (~y, ~') for every (x', y) C E' x F . Moreover,

I1~11 < I1~111

320

1. Banach Spaces

Let ((x'~, y~))~e, be a finite family in E' x F such that <.,,'~>y~ - o. tCI

We show t h a t Z(~Y~,

x:) - o.

tel

Since the m a p

F

>IK,

y,

>(uy, x'}

is linear for every x' C E ' , we may assume (yt)~el to be linearly independent. Since

tel

for every x E E , it follows

(x, z:) - 0 for every x C E and t C I . Hence x 't- 0

for every t C I

and

<~y,, ~:> - o. t(SI

By the above proof (and Corollary 1.3.3.4), the map

g's

, IK,

~(.,x:)y~, tg=]

, E( u y ~,x : ) tC I

is well-defined; it is obviously linear. Let ((x", Y~,))AeL be a family in E" x F' such that

AEL

and

ACL

for every y E F . Let ((ate, Y~))~cI be a finite family in E' • F . Then

1.6 Nuclear Maps

rE[

)tEL

)tEL

,kEL

t,EI

LEI

)tEL

)tEL

)tEL

)tEL

)tEL

~EL

tEI

321

tel

tEI

tEI

tEI

tEI

)tEL

by Corollary 1.3.6.6 and Theorem 1.3.4.2 b). Hence ~ E s

tel

F)' and

I1~1~ ~11~"1 y,~ll. )tEL

It follows that

I~1 ~ I1~111.

m

322

1. Banach Spaces

1.6.2 Examples

Example 1.6.2.1

Take IK 2 with the Euclidean norm. Take u E/2(IK 2) and

let

g be the matrix associated to u . Let A 2 .= lal 2 + 1/319.+ 1712 + 1512, and

a := I ~ -

~1.

Then Ilul[ 2 = A 2 + 2 A . A s s u m e that A r 0 and choose 0 C IR such that oz(~- g 7 = A e i~ Let v be the operator on IK 2 associated to the matrix

vIA 2 + 2 A

~-

t3e -iO

-~ + cte -iO

Then

I~11= 1 and the following are equivalent for every w E s a)

W :

b)

If ~

#

?2.

denotes the element of s

defined in Theorem 1.6.1.12 a),

then

~(~) = Jl~ll,

Case 1

A = 0

In this case Im u is o n e - d i m e n s i o n a l and

1.6 Nuclear Maps

Ilul~=liu[[ 2 - ~

323

1( A 2 + v / A 4 - 4 A 2 )=A ~= A"+ 2 A

(Corollary 1.6.1.4, Example 1.2.2.8). Case 2

A :/_ 0

By Theorem 1.6.1.12 b) (and Corollary 1.3.3.8

2) is identified with s

1.2.1.6),

Ilwll- 1},

IlUlll- sup {re(u, w) lw E s where s

b), Corollary

Take w E s

2) with IIw[I = 1, and

let

c d by the matrix associated to w. Set

A~:;[al ~+lbl ~+lcl ~+ldL 2,

D-ad-bc.

Then 1

(Example 1.2.2.8) and this is equivalent to A 2 - IDI 2 = 1. Since (u, w) = a a +/3b + "yc + 6d, we have to find the supremum of 1 (c~a + ~b + 7c + 5d + -&---d+ -~ + ~ + 5d) re (c~a + ~b + 7c + 5d) = -~

in the variables a, b, c, d on

A2 - [DI2= 1. We first show that the supremum is not attained at a regular point (a, b, c, d) of A 2 - [D[ 2 = 1. Assume the contrary. Then there is a A c 1R with

324

1. Banach @aces

A

a - ~

O(A 2 - IDI 2) Oa = A(g- Dd),

A O(A 2 - IDI 2) = A(g + ~c), i)b

/3-- -~

IDI 2)

A O(A 2 -

7= 2

Oc

A__ O(A 2 -

_

2

= A(-d + D b ) ,

ID 2) = ~X(-d--Da)

Od

and we get the contradiction that 0 7s a5 - / 3 7 - A 2 ( a d -

Dial 2 - -Did[ 2 + - D 2 a d - b c - -Dlb[ 2 - -D[c 2 - -D2bc) -

=A2(D-DA2+D2D)-A2D(1-A2+[D[

2) - 0 .

Hence the supremum is attained at a singular point (a, b, c, d) of A 2 - [D[ 2 - 1. Now 0

0

O(A 2 - I D 2)

-

Oa

-

O(A2

-

=-~-Dd,

IDI2) = b + ~ c ,

Ob 0

-

0 =

O(A2-lD[2) Oc

=-d+Db,

O(A2

= -d-

-

[DI2)

Da,

Od

so that D - ad-

bc - -~2 ( a d -

bc) - -DID[ 2

Since obviously D r O, ID[

=

1,

A 2 -

1 + IDI 2 - 2.

1.6 Nuclear Maps

325

Let

D = e i~ w i t h -1- r IR. T h e n

- d - - b e -iv

--d- ae -i~-

lal ~ § Ibl ~ -- 1A2

-

2

1.

H e n c e we have to find t h e s u p r e m u m of M -

re ( a a + ~b + 7c + 3d) -

1 (cm + fib + 7c + 3d + ~ -~

_ 1_ ( a a + gb - 7-be i~- + (~-aei'r + ~ 2

-nt- gb

-

-

+ / 3 b + ~-~ + ~--d) =

-~be -iT --~ -Sae -i~)

on

la 2 + ]b 2 _ 1. T h e r e is a A E ]R so t h a t

~X~-- ~X0(la12 + Ib12) - 2 0 M Oa

~

- a + -Se -i~

) ( b - A O([a[2 + Ib12) OM Ob = 2-5-~- - 9 -

~e_~

9

Hence

~ - ~2(lal = + Ib =) - I ~

= + I~1= Jr 2re ~ e ~ Jr I~ = + I~ = - 2re 9 ~ e ~ -

= A 2 + 2re A e i(~

A M : re ( a ( ~ + 5e i~) + / 3 ( - ~ -

= re (1~1 ~ + ~

,

7e i~) - 7(/3 - ~ e - i ~ ) e i~ + 5 ( a + 3 e - i ~ ) e i~) :

+ IZl: - Z <

~ - ~ 9 ~ ~ + I~l: + ~-~'~ + I~1 ~) =

= re (A 2 + 2 A e i(O+'r)) -- ,~2,

326

1. Banach Spaces

M-A. Hence T = --0.

I1~11~: l

[ a + Se-i~ ~ _ fle_~o

w -- ( A2 + 2A)2'Proposition

A2 + 2A,

1.6.2.2 ( 1 ) ( 3 )

fl-'Te-w 1 ~ + ae_~ o = v.

Let S , T be sets and take

p E

II

[1, oc]U{O}.

Given k C fl'P(T, S), set u

u

k. e~(T) ~

e~(S),

x,

>kz

(Proposition 1.2.3.8 a) ). Then u

u

u

Ilk lco(Z)ll, = I kll, = I kll

k c Z,'(e~(T),eP(S)),

for every k E fl'p(T,S) and the map

e"p(T,S)

, Z:'(~o(T),eP(S)),

k

~, ~lco(T)

is an isometry.

Step 1

Ca(co(r),

3!k e t~I'p(T, S), o

k I ~o(T) - u, Ilkll _< It-ll,.

Let ((x',, Y,)),c, be a family in co(T)' x gP(S) such that

Y~ IIXII ly, I < o~ and

tCI

for every x c co(T). We identify co(T)' canonically with el(T) (Example 1.2.2.3 e) ) and set

k.TxS

~:,

(t,~),

>(~r

1.6 Nuclear Maps

327

Let q be the conjugate exponent of p. Take t E T . T h e n

sES

sES

tel

sES

sES

_<

tel

tel

sES

~ Ixt(t)l Ily~llIly'll~ = Ily'll~~ Ixt(t)llly~ll LEI

tEI

for every y' E IK (s) so t h a t k(t, .) c eP(S) and

Ilk(t,-)11, _< ~ Ix[(t) Ily~ll tEI

(Proposition 1.2.2.2). Now

E rl~(,, )ll~ _~ E ( E i~:(t), IJ~,l,) tcT

tCT

=E

LEI

~ l l ( E ix:(,) ) = E I~,l ,l~:l.

LCI

tET

tEI

Hence k E gl'P(T, S) and

Ilkll _< I1~111. Given t E T , u

k~,

T

k(t, .) -

-

ueT,

so t h a t u

kx

~

~tx

for every x C IK (T) Since IK (T) is dense in co(T) (Proposition 1.1.2.6 c)), u

klco(T) - u . T h e uniqueness of k is obvious.

Step 2

k C ~I'P(T, S) ==v

{~ ~ ~'(I~(T), e~(s)), u

IIkl~o(r)ll,

u

-Ilkl,

-

Ikll.

328

1. Banach Spaces

Given t 9 T, take X ,t . ~oo (T) -------+]Z,

Yt

Then x t 9

(T)' and Y t 9

x,

)x(t),

"= k ( t , . ) .

(S) for every t 9

I1~'r Ily, ll~ - I l k

<

oo,

t6T

and

;x - Z

k(t, )~(t) - ~ ( x ,

t6T

~',)y~

t6T

for every x 9 g~176 Hence u

k c s176176

e~(s))

and (_J

tll~_< Iltl.

By Step 1 (and Proposition 1.6.1.5), u

u

I[kl _< Ilk co(T)lll < I[kl[1. Hence

u

u

I k (:o(Z) ~ - I l k I~ Corollary 1.6.2.3

Ikl

m

Let S, T be sets. Take p 9 [1, oc[ and let q be the conju-

gate exponent of p.

a) (h(t, s)(ueT)(s))(t,s)er• U 9 s

is summable for every h 9 g~'q(T,S) and

eP(S)). Given h 9 e ~ 1 7 6

let

(t,s)6TxS b) h e ~ l ( c 0 ( T ) ,

is an isometry.

~P(S))t fop every h 9 ~.~

and the map

1.6 Nuclear Maps

329

By Proposition 1.6.2.2, the map u

u

xJ

k " co(T) --+ gP(S),

) kx

is in /21(c0(T), eP(S)) for every k 9 e"P(T,S) and the map u

gl,p(T, S)

, Cl(c0(T),

eP(S)) ,

k l

)k

is an isometry. We identify gl,p(T,S) with s gP(S)) by means of this isometry. By Proposition 1.2.3.6, (h(t, s)k(t, s))(t,s)~TxS is summable for every k 9 el,p(T, S) and h 9 g~,q(T, S ) , the map > IK ,

h" gl'p(T,S)

k,

>

~

h(t, s)k(t, s)

(t,s)CT•

belongs to gl'p(T, S)' for every h e

e~,'(T, S), and the map

> el'p(T, S)',

g~'q(T, S)

h,

)h

is an isometry. This proves the corollary. P r o p o s i t i o n 1.6.2.4 ( 1 ) ( 3 ) exponents. Given k C gl,q(s, T), let A

k" gP(T)

II

Let S , T be sets and let p,q be conjugate

~ g~(S),

N

x,

> kx

(Proposition 1.2.3.4 a) ). N

~) If p # ~ ,

th~n k c

L~(e~(T),eI(S)) fo~

g~'q(S,T)

-~/:I(~P(T), ~1(S)),

~y

k e e~,~(Z,T) and th~

map n

k,

)k

is an isometry.

b) If p -

A

cxD, then k E / : I ( g ~ ( T ) , el(S)), N

n

Ilklco(T)lll = IIk]l~ - I kll for every k E gl,l(S, T) and the map

e 1'1(S, T) is an isometry.

> ~1 (C0(T), g1(S)),

k,

0

> k lco(T)

330

1. Banach Spaces

Put E:=

~ eP(T)

t

if if

co(T)

pr p = cx~.

{ 3k E ~"q(S,T), Step 1

u E s

==~

n

klE = ~,

Ilkll _< I1~ I1.

Let (( x'L, YL))~I be a family in E' • gt(S) such t h a t

IIx: I Ily~ll < c~ LEI

and

eEI

for every x e E . Identify E' with

k. SxT

e~(T)

~IK,

(Example 1.2.2.3 d), e)) and set

(~,t),

~(~r

Then

tET

tCT

for every x E IK (T) and s C S. By Proposition 1.2.2.2,

LCI

for every s C S . Hence

k Es

T)

and

Ilkll _< l u l l .

k(s, .) C gq(T)

and

1.6 Nuclear Maps

331

We have n

k~T - k(., t) : ~ T

for every t E T and so N kx

for every x

E

IK (T) .

Since

:

ttX

is dense in E (Proposition 1.1.2.6 c)),

]K (T)

M

klE=~. The uniqueness of k is obvious. {k Step 2

k E ~I'q(S,T)

==~

Cs n

n

IlklEIl~ = Ilkll, = Ilkll.

Given s c S, let x s' : = k ( s , - ) ,

ys:=esS.

Then x 's C gP(T)' and ys c t~1(S) for every s E S,

IIx'~ll Ily~ll = ~ s6S

IIk(~, )llq = I kll <

s6S

(Example 1.2.2.3 b)), and

( sES

sCS

)

s

n

tET

s

~

kx

sES

for every x e ~P(T) (Example 1.1.6.16). Hence

o

(

k e 121 t~P(T), t~l(S)

)

and n

I klll <~ Ilkll 9 By Step 1 (and Proposition 1.6.1.5), n n l l k l l - ~ I}k I ElI1 -~ Ilklll-

Thus

n

n

IiklEill-Iiklll--IIkll .

I

332

1. Banach Spaces

Corollary 1.6.2.5

( 1 ) (3)

Let S , T be sets. Take p C]l, cc[ and let

q be the conjugate exponent of p.

a) (h(s, t)(ueT)(s))(s,t)~S• is summable for every h e t~'q(s, T) and u 9 ~..'(~q(T),~I(S)). Given h e ~c~'q(S,T), let h" s

~, IK,

b) h c s

u, ~ ~

E h(s't)(ueT)(s)" (s,t)CSxT

gx(s))' for every h C g~176 t'~'q(S,T)

~s

and the map

',

h,

>h

is an isometry.

By Proposition 1.6.2.4, the map N

is in s

M

~1(S)) for every k C gl'P(S,T) and the map el'P(S, T)

~ ~l(eq(T), el(S)),

is an isometry. We identify gl,P(S,T) with ~s By Proposition 1.2.3.6, (h(s, t)k(s, t))(s,t)eS• k 9 t~I'p(S, T) and h e t~~176 T), the map h" el'P(S, T) ~

IZ ,

k,

M

k,

>k

via this isometry. is summable for every

> E h(s, t)k(s, t) (s,t)ESxT

belongs to e 1';(S, T)' for every h E e~176(S, T), and the map g~'q(s,r)

~ gl'p(S,T)',

h,

is an isometry. This proves the corollary. Example 1.6.2.6

~h I

( 3 ) Let (a)Lci be an absolutely summable family in the

Banach space E and define u " g~176

; E,

x,

tCI Then u is nuclear and

Ilulll <_ Z IJajl tCI

1.6 Nuclear Maps

333

Given ~ E I , define

~,.e~(z)

>~,

~,

>~(~).

Then

~: ,,x:j,I,a~l,- ~ I,a~J, tel

tel

and

tEI

for every x E f ~ ( I ) .

I

334

1. Banach Spaces

1.7 Ordered

Banach

spaces

C*-algebras are equipped with an intrinsic order relation with respect to which they become ordered Banach spaces. Properties of ordered Banach spaces which are used later in the book are presented in this section. Because of its importance in the later study of W*-algebras, order continuity is also studied here. 1.7.1 O r d e r e d n o r m e d spaces

Definition 1.7.1.1

( 0 )

An ordered v e c t o r space is a vector space E

with an order relation <_ such that given any x, y, z C E and a C IR+ x + z <_ y + z and crx < t~y whenever x < y . Defining

E+ := {x E E I x >_ 0}, we call the elements of E+ p o s i t i v e and the elements of - E +

P r o p o s i t i o n 1.7.1.2

negative.

( 0 ) I f E is an ordered vector space then E+ is a

sharp convex cone in E .

Given x,y E E+ and ~ E IR+, x + y, ax E E+,

so that E+ is a convex cone in E. From x C E+ N ( - E + ) , it follows successively that -xCE+,

0<-x,

x_<0_<x,

x=0,

i.e. E+ is sharp. P r o p o s i t i o n 1.7.1.3 cone in E .

I ( 0 ) Let E be a v e c t o r space and C a sharp convex

Then there is a unique order relation <_ on E ,

which E is an ordered vector space with

E-.~ ---C.

with respect to

I. 7 Ordered B a n a c h spaces

335

Define the relation < by x
for x , y c E . Then, given x , y , z C E and a E IR+, x-x-OcC,

x<_x,

x < _ y , y <_ z =:::~ y - x ,

z-

z-y

~ = ( z - y) + ( y -

~) c c ~

x <_y, y ~_x==:::vy-x,

---~ y -

x C C M (-C)

==:~ y -

x-y

x E C ==~ t ~ y - t ~ x

x c C v==:v x - O

~ < z,

c C ::==~

x = O ===~x = y ,

x ~_y==~ (y+z)-(x+z)=y-xC

x ~ y ::~ y-

E C ===>

C ~

- a (y -

x+z

x) c C ~

<_ y + z ,

a x ~ t~y,

c C c==~ O < x .

Hence < is an order relation on E with respect to which E is an ordered vector space such that E+ -

C.

Let ~ be an order relation on E which renders E an ordered vector space with E§ -

C.

Then x -~ y v:==:>O ~_ y -

x c::~ y-

x E C v:::=> x ~_ y

for every x, y E E and so _< coincides with ~ .

I

336

1. Banach Spaces

[

Definition

1.7.1.4

\

( 0 ) An ordered n o r m e d space is an ordered vector

space E endowed with a norm with respect to which E+ is closed. In this case we put

E+# := E # M E+. An ordered B a n a c h space is an ordered normed space with a complete norm.

If T is a topological space, then C(T) endowed with the usual order is an ordered Banach space. P r o p o s i t i o n 1.7.1.5 ( 0 ) Let E be an ordered normed space and let (xt)t~i, (Yt)te, be families in E such that xt <_ Yt for every t E I . a) If ~ is a filter on I such that limxt and limy~ exist, then t,~

t,~

lim x t < lim y~. t,~

--

t,~

b) If (xt)te, and (yt)tc, are summable then

C)

If

Ext

<_ E y e .

tel

tel

(Zt)tE I i8 a summable family in E+ then 0<_ ~-~ zt <_ ~-~ z~ c tCJ

E+

tel

for every finite subset J of I .

a) limt,;~y~ -limt,;~xt = 1},~n(yt- xt) C E+. b) By a), Ex~-limEx,~_limEyt-Eyt. tEI

tEJ

tEJ

tCI

c) follows from b). Corollary 1.7.1.6

( 0 ) Let E be an ordered Banach space such that 9, y e E + , ~

< y ~

Ilzll _< Ilyll

a) For all x , y C E , - y _< x _< y ~

IIxll < 311yll.

1.7 Ordered Banach spaces

b)

If

337

(XL)eEI, (Yr)rEI are famlies in E such that (Y~)~e, is summable and -y~ <_ x~ <_ y~

for every ~ E I then (X~)~EI iS also summable and

~EI

eEI

~EI

From O<x+y<2y

it follows

I1~11 = I1~ + y-

yll < IIx + yll + Ilyll < 1]2yll + Ilyll = 311yll

b) Take c > 0. There is a J E ~3/(I) such that y~ < for every K E q 3 i ( I \ J ) (Proposition 1.1.6.6). By a),

for every K E ~ / ( I \ J ) . Proposition 1.7.1.5 b),

By Proposition 1.1.6.6), (x~)~eI is summable and by

~EI

Corollary 1.7.1.7 space such that

( 0 )

tEI

LEI

Let E be a finite-dimensional ordered Banach

x,y E E§

x < y ~

Then there is an c~ E JR+ such that

for every summable family (x~)~eI in E+.

IJzll < IlYll-

338

1. Banach Spaces

Assume the contrary. Then for every n E IN there is a finite family (xn,~)~eIn in E+ such that

~~ < v '

Z II~o,~ll>_1. rein

Consider the family (xn,~)ne~,~eI,. Take c > 0 and no E IN such that oo

1

~~<~ n:n

0

Let J be a finite subset of {(n,t)in > n0,L E In} and put p'-

sup n. (n,t)EJ

By Proposition 1.7.1.5 c), P

0_< E ~o,~-< E E~o,, (n,~)EJ

n=no rein

]m~

By our hypothesis about E ,

_<

Xn,t n=no rein

x~,~ < n~---n 0

n----n 0

1

By Proposition 1.1.6.6, the family (Xn,~)neIN,LcX,~ is summable. By Proposition 1.1.6.14, the family (Xn,L)neIN,Lel, is absolutely summable, and we get the contradiction

- E E llxn,II< nEIN rein

C o r o l l a r y 1.7.1.8

( 0 )

~

m

Let x' be a linear form on the ordered Banach

space E such that x ~( E+ ) C IR+ .

a) x'iE+ is continuous at O. b) If the absolutely convex hull of E#+ is a O-neighbourhood in E , then x' is continuous. a) Assume the contrary. Then for each n E IN there is an xn E E+ with 1

II~ll ___ ~ Put

,

X ! (x~)

___ ~.

1.7 Ordered Banach spaces

339

X "--- ~ X n nc IN

(Corollary 1.1.6.10 a =v c). T h e n P

~xn

<_x

n=l

(Proposition 1.7.1.5 c), so t h a t

_<

x,(s

n----1

<

n--1

for every p 9 IN and this is a contradiction. b) By a), x' is bounded on E+e . Therefore it is bounded on the absolutely convex hull of E+e . Hence x' is b o u n d e d on a 0-neighbourhood in E and so it must be continuous. Proposition

1.7.1.9

I

Let E, F be ordered normed spaces such that the vector

subspace of E generated by E+ is dense in E . Then there is a unique order relation on Z:(E, F) with respect to which I:(E, F) is an ordered vector space and s s

- {u 9 f~(E,F) I u(E+) C F+} .

is closed in the topology

of pointwise convergence and so

/2(E,F)

is an ordered normed space. The elements o f / : ( E , IK)+ will be called positive linear f o r m s on E . Put jc := {u e L ( E , F ) Iu(E+) c F + } . is clearly a convex cone in I:(E, F ) . Take u E 9~" n (--.~').

Then

~(E+) c r+ n ( - F + ) = {0}. Thus u vanishes on the vector subspace of E generated by E + . By hypothesis, u - 0 and ~ is sharp. The first claim now follows from Proposition 1.7.1.3. The second claim is easy to prove.

I

340

1. Banach Spaces

1.7.2 O r d e r C o n t i n u i t y [

\

( 0 ) Let A be a subset of the ordered set E . An upper

Definition 1.7.2.1

(lower) bound f o r A in E is an element x 9 E such that y < x

(x < y)

for every y 9 A . If there is a least upper bound (greatest lower bound) for A in E then it is called s u p r e m u m ( i n f i m u m ) of A in E and it is denoted by E

E

Y x or V x (by A x or A x ) . I f A = { x , y } , xcA

xEA

xEA

then we put

xEA E

xVy'=xVy:=

V z,

zEA

E

xAy'--xAy:=

A z.

zEA

E is called complete if every nonempty upward (downward) directed set of E , which is bounded above (below) has a supremum (infimum) in E .

E is

called a - c o m p l e t e if the previous conditions are fulfilled for countable subsets of E . If in addition E is a normed space, then we will use the expression order complete and order a - c o m p l e t e instead of complete and a-complete, respectively, in order to avoid confusion. Let (x~)Lci be a family in E and put A "= {x~ I ~ 9 I } . We extend the above notions for (xL)~ei by replacing it with A and define E

tel

tel

E

= Vy, yEA

tel

E

E

t@I

yEA

E is called a lattice (Dedekind, 1897) if any two elements of E have both a supremum and an infimum (or equivalently, if every finite subset of E has both a supremum and an infimum). A vector lattice is an ordered vector space which is a lattice. A band of an order complete vector lattice E is a vector subspace F of E such that: E

1) x, y c F ~ x v y c F . 2) x c E ,

yCF,

O<_x
3) If (xt)te, is an upward directed family in F then V xt belongs to F if tEI

it exists.

1.7 Ordered Banach spaces

Remark.

341

An ordered set E is a-complete iff every bounded increasing (decre-

asing) sequence in E has a supremum (infimum). P r o p o s i t i o n 1.7.2.2

( 0 )

Let E be an ordered Hausdorff vector space

such that E+ is closed and that the translations and multiplications with - 1 are continuous (for example let E be an ordered normed space). Let A be a nonempty upward directed set of E , ~ the upper section filter of A , and x a point of adherence of ~ . Then x is the supremum of A .

Take y e A. Then {z 9 A i z > y} 9 ~ so that x e {z 9 A I z

x-

y c {z-y

>y},

l z c A , z >_ y} c E+ = E + ,

y<_x.

Thus x is an upper bound for A. Let y be an upper bound for A. Then y-zGE+

for every z C A and so y-x

e {y-z

I z 9 A} C E+ -- E+.

x
Hence x is the supremum of A. D e f i n i t i o n 1.7.2.3

I

( 0 ) Let u" E --+ F be a linear map from the ordered

vector space E to the normed space F . Let 91 be the set of all downward directed (countable) sets of E whose infimum is O. For each A E 92 , let 2)A denote the lower section filter of A . Then u is said to be order continuous (order a - c o n t i n u o u s ) if u(Y)A) converges to 0 for each A E 91 . The vector subspace of IK E generated by the order continuous (order or-continuous) positive linear forms on E is denoted by E ~ (resp. E ~).

Every linear form of E ~ (of E ~ ) is order continuous (order a-continuous).

342

1. Banach Spaces

P r o p o s i t i o n 1.7.2.4

( 0 ) Let E be an ordered Banach space, F a normed

space, and u" E -+ F an order a-continuous linear map.

a) ulE + is continuous at O. b) If the absolutely convex hull of E#+ is a O-neighbourhood in E , then u is continuous. a) Assume that u[E+ is not continuous at 0. Then for every n C IN there is an xn C E+ with

IIx,~ll _<

2~,

II~x.II _> 1.

Given p E IN, put OG

yp'- y~xn rt--p

(Corollary 1.1.6.10 a =~ c). Then, for p C IN,

Ilypll ~ ~ 1 x.ll < ~ 2n--p

=

2p-l'

n=p

1 <_ Iluxpll = I l u y p - uYp+lll < Iluypll + [luyp+lll.

By Proposition 1.7.2.2

pEIN

n=l

SO

AYp-0.

pEIN

Since u is order a-continuous, (uyp)pe~ converges to 0. This contradicts the above inequalities. Hence ulE + is continuous at 0. b) By a), u is bounded on E+# . By the hypothesis of b), u is bounded on a 0-neighbourhood in E . Hence u is continuous. 1 P r o p o s i t i o n 1.7.2.5

( 0 ) Let E be an ordered Banach space such that

IIx + vii = IIxll + IJvll for every x, y E E+ . Let A be a nonempty upward directed subset of E+. Then the following are equivalent:

1.7 Ordered Banach spaces

a)

A is bounded above.

b)

A is bounded in norm.

c)

The upper section filter of A converges.

d)

There is a point of adherence for the upper section filter of A .

333

e) A has a supremum. If these conditions are fulfilled, then the upper section filter of A converges to the supremum of A . In particular, E is order complete and every operator defined on E is order continuous. a =:> b. Let y be an u p p e r b o u n d of A . By hypothesis

Ilzll _< II~Jl for every x C A , so t h a t

sup Ilxll ~< Ilyll 9 xCA

b ~ c. We set II

II

xEA

Take a C A and suppose x, y c A , a < x , a < y. Since A is upward directed, there is a z C A with

x<_z,

y<_z.

Then

IIx - yll ~< IIx - zll + IIz - yll = Ilzll- I xll + Izll- Ilyll ~< 2(~ -Ilall) It follows t h a t the upper section filter of A is a Cauchy filter and it therefore converges. d :=> e and the last assertion follow from Proposition 1.7.2.2. c ==> d and e =:~ a are trivial, Proposition

1.7.2.6

l

( 0 ) Let E be a normed space. A s s u m e that E' is an

ordered vector space such that E'+ is closed in E'E . Then every norm bounded upward directed nonempty set A' of E'+ has a supremum x' and the upper section filter of A' converges to x' in E'E .

344

1. Banach Spaces

We may assume that A' C E '# . Since E'#E is compact (Alaoglu-Bourbaki Theorem), the upper section filter ~ of A' has a point of adherence x' in EE#,. By Proposition 1.7.2.2, x' is the s u p r e m u m of A'. In particular, x r is the unique point of adherence of ~ in EE#,. Using once again the fact that EE#, is compact, we deduce that ~ converges to x r in E ~ . Proposition

i

Let E be an ordered normed space whose norm is mo-

1.7.2.7

notonic increasing (i.e.

Ixll << IlYll whenever x , y 9 E+ and x <
a normed space and define .7z by .T" := {u 9 s

F) i u is order continuous (order a-continuous)}.

Then .7: is a closed vector subspace of s

F).

It is easy to see that 9v is a vector subspace of E(E, F ) . Take u 9 ~ and let A be a downward directed (countable) subset of E with infimum 0. Take > 0 and x0 9 A. There is a v 9 ~ such that

II~- vii <

C

2(1 + il~01i)

Further there is an x 9 A such that C

for every y 9 A with y _< x. Then C

fi~yil _< li(~- v)yi + ilvyi _< 2(1 + llx011) for y 9 A with y _< x and y _< x0. Hence u(r

converges to 0 and u 9 9r . i

Example

1.7.2.8

Let T be a set and take p C {0}U[1, cx~].

a) There is a unique order relation on ~ ( T )

such that ~P(T) is an ordered

vector space and

e~(T)+ = {~ c e~(T) I ~(T) c ~+} b) ~P(T) is an order complete vector lattice and an ordered Banach space. c) Every order g-continuous linear map of ~P(T) into a normed space is

continuous.

IIx -~- YlI1 ----IIXlll -~- IlYlII"

1.7 Ordered Banach spaces

3~5

a) follows from Proposition 1.7.1.3. b) and d) are easy to see. m

c) follows from Proposition 1.7.2.4 b). E x a m p l e 1.7.2.9

Let T be a set. Take p c {0}U[1, oo], and let q be the

conjugate exponent of p. Given x E/~q(T), define ~'gP(T)

) IK,

Y'

>E x ( t ) y ( t ) tET

(Example 1.2.2.3 a)). a) Given x C gq(T), the linear form ~ is positive iff x is positive (Example 1.7.2.8 a)). b) x e eP(T) 7r, whenever x e eq(T). c)

gP(T) ~ = (x' l x' is an order continuous linear form on gP(T)} ,

~P(T) ~ - (x' l x' is an order a-continuous linear form on gP(T)}. d) The map ~ . q ( T ) ----} ~ . P ( T ) 7r ,

x,

) 'x

is an isometry of ordered Banach spaces (Proposition 1.7.1.9, Proposition 1.7.2.7). a) is easy to see. b) Since every element of t~q(T) is a linear combination of positive elements it is sufficient to show that ~ is order continuous for every x C t~q(T). Case 1

p = 1

The assertion follows from the last assertion of Proposition 1.7.2.5 and Example 1.7.2.8 d)). Case 2

p :/: 1

It is obvious that ~ is order continuous if x C IK (T) . Since IK (T) is a dense set of gq(T) (Proposition 1.1.2.6 c)), it follows from Proposition 1.7.2.7 (and Example 1.2.2.3 b)), that ~ is order continuous for x C gq(T). c) follows from the fact that gP(T) is a vector lattice. d)

Case 1

p -~-

By Example 1.2.2.3 d), the map

346

I. Banach Spaces

gq(T) , > e ( T ) ' ,

x,

>

is an isometry of Banach spaces and so the assertion follows from a) and b). Case 2

p = c~

Let x' be a positive order continuous linear form on g ~ ( T ) . We define

x" T

>IK,

t,

>x'(et) .

Then

x(t) - ~ tEA

x'(et) = x(ea) <_ x(eT)

tEA

for every d E ~ s ( T ) and so x E e l ( T ) . Take y E g~~

Then

{Yer\A ]A E gts(T) } is a downward directed set of g~(T) with infimum O. Thus 0 =

inf

AE~I(T)

x'(y)-

x'(yer\A) = x ' ( y ) -

sup

x'(yeA)=

AEq3I(T)

sup

AE~s(T )

x(yeA),

x(yeA)=X(y).

sup AEq3I(T)

Since y is arbitrary and every element of t ~ ( T )

is a linear combination of

positive elements of t ~ ( T ) , we see that x' - 5. Hence the map

eX(T)

> e ~ ( T ) ~,

x,

>

is surjective. By a) and Example 1.2.2.3 b), the above map is an isometry of ordered Banach spaces.

Remark.

I

Let ~ be a free ultrafilter on T closed under countable intersections.

Then the linear map

e~176

~ IK,

x,

~lim x(~:)

is positive and order a-continuous but not order continuous. Let Wl be the first uncountable ordinal number endowed with the usual topology and let ~ be the filter on wl of the compelements of the countable subsets of wl. Then the map C(wl) ----4 IK,

x,

~lim x(~)

is positive and order a-continuous but not order continuous.

1.7 Ordered Banach spaces

Proposition

1.7.2.10

Let E

347

be a a - c o m p l e t e ordered vector space. The

countable f a m i l y (xt)~ei in E+ is called o r d e r s u m m a b l e if

tEJ

is bounded above. In this case we define

~-x~

:= V ( ~

eel

x~ I J ~ ~ : ( 1 ) } tEJ

Then, given a linear f o r m x ~ on E , the following are equivalent: a) x' is order a - c o n t i n u o u s . b) given an order s u m m a b l e countable f a m i l y

(x,)tei

in E + , the f a m i l y

(x'(x~))tei is summable and

~EI

LEI

a ==> b. We m a y assume I to be infinite. Let ~ 9IN --+ I be a bijective m a p . Then

<

- Xt

--

Xr

nEIN

is a decreasing sequence in E with infimum 0.

Thus

0

-

mx'(Z

x~-

x~(k)

x~

?2--+00

~EI

k=l

-

lim n-+oo

~EI

k=l

x~(k)

)

,

k=l

LEI

and the assertion follows from P r o p o s i t i o n 1.1.6.14 c => a. b ==v a. Let A be a downward directed countable set of E with infimum 0 and let ~ be its lower section filter. A s s u m e t h a t x ' ( ~ ) d o e s not converge to 0. T h e n there is an c > 0 such t h a t for every x E A there is some y E A such that y<x

and

Ix'(y) l > ~. Thus there is a decreasing sequence (Yn)nE~ in A with infimum 0 such t h a t

I~'(y~)l > for every n E IN. Given n E IN, define

Xn .m Yn -- Yn+l 9

338

1. Banach Spaces

Then

yn ~<m

Xk

k E IN k>n

for every n E IN. By b), =

kEIN

k >_ n

for every n E IN. We deduce the contradiction that 0-n~o~lim I E x'(xk)l = ~-,~limIx'(Yn)l > ~ .

I

kEIN

k >_ n

P r o p o s i t i o n 1.7.2.11 Let E be a a-complete ordered vector space and (x~)ne~ a sequence of order continuous (order a-continuous) linear forms on E such that (x~(x)),e~ converges for every x E E . Then x''E--~IK,

x:

!

; n---~ lim(:X3 Xn(X )

is order continuous (order a-continuous).

Let (X~),EI be an order summable countable family in E+. Given a linear form y' on E , define y" I

) IK,

to

) y'(x~) .

Let (aL)~EI be a bounded family in JR+. By Proposition 1.7.2 10 a =~ b x n E t~l (I) and

for every n E IN. Hence lim E

o~x'~(~)

n-~oc

LEI

--

x' ( E - - - )OZ~XL

"

LEI

It follows that (X~)nEr~ is a weak Cauchy sequence in gl(I). By Schur's Theorem (Theorem 1.3.6.11), x ' E gl(I) and lim IIx" - x'll, - 0.

n--.+oo

1.7 Ordered Banach spaces

Thus

f

349

)

Xt k tEI

~EI

tGI

tEI

By Proposition 1.7.2.10 b => a, x' is order a-continuous. Now suppose t h a t each x 'n (n C IN) is order-continuous and t h a t x' is not order continuous. Then there are a downward directed set A of E with infimum 0 and an c > 0 such t h a t for every x E A, there is a y E A with y < x and

Ix'(y)l > ~. We may construct a decreasing sequence (xn)ne~ in A inductively such that for every n C IN I X t ( X n ) I ~> -C

and k ~ ~

1

~

tx~(x~) I _< - . n

We put X:~-

AXn. nE ]N

Then !

!

Xk(X ) -

lim Xk(X,~ ) - 0

n--~ OQ

for every k C IN and I x ' ( x ) l - lim Ix'(xn)l _> e. n---9.oo

This leads to the contradiction t h a t

< x'(x)l-

lim Ix~(x)l = 0.

I

k--+c~ /

D e f i n i t i o n 1.7.2.12

(

0

)

Let T be a locally compact space. The open set

U of T is called an e x a c t set o f T if it is of the f o r m

u-{~r f o r some x E C(T).

350

1. Banach Spaces

Let if; be the a-algebra on T generated by the exact sets of T . The elements of T, are called Baire sets and the T,-measurable functions on T are called Baire functions. T

is called a Stone space ( a - S t o n e space) if the closure o.f any open

(exact) set of T is open. A hyperstonian space is a Stone space T in which

U

supp#

ttECo(T)~

is dense. By Urysohn's Theorem, every open a - c o m p a c t set of T is exact. The intersection of a finite family of exact sets is exact, as is the union of a countable family of exact sets. If T is metrizable, then every open set of T is exact, so that every Borel function on T is a Baire function. Proposition

1.7.2.13

( 0 )

Let T be a locally compact space. Then the

following are equivalent: a)

T is a Stone (a-Stone) space.

b) If x is a bounded Borel (Baire) function on T , then there is a y E C(T) such that {x ~ y} is meager. c) Every nonempty (countable) family (x~)~e, in C(T)+ has an infimum y in C(T) and

{t E T ly(t ) ~ infx~(t)} tEI

is meager. d)

C(T) is order complete (order a-complete).

e)

Co(T) is order complete (order a-complete).

The function y in b) is unique and

y(T) c ~(T). a =~ b. Let ~R be the set of subsets A of T for which there is a clopen set U of T such that

(A\U) U(U\A) is meager. By a), 9~ contains the open (exact) sets of T . It is easy to see that ~R is a a-algebra. Hence, every Borel (Baire) set of T belongs to 9~. By the definition of 9~, for each A E 9~ there is some y E C(T) for which {CA -~ y} is meager. It follows that b) holds for step functions on T with respect

1.7 Ordered Banach spaces

351

to 9~. Since x is a bounded Borel (Baire) function on T , there is a sequence ( X n ) n ~ of step functions on T with respect to 9t: converging uniformly to x . For every n C IN, there is a Yn C C(T) fbr which {xn ~ Yn} is meager. Since T is a Baire space,

Ilym - y,~llo~ < IIx,,, - xnlloo for m, n C IN. Hence (Yn)nEIN is a Cauchy sequence in C(T). Set y "-- lim yn. n--). oo

Then

{X # y} C U {xn # Yn}. nE]N Hence {x # y} is meager. b ==> c. Define y'T

>r

t,

>infx,(t).

y is a bounded Borel (Baire) function. By b), there is an x E C(T) for which {x # y} is meager. Since T is a Baire space, x is the infimum in C(T) of c :=> d =::> e is easy to see. e ==~ a. Let U be an open (exact) set of T . We are required to prove that U is open. We may assume U to be relatively compact. First assume that U is open and put

.T" "- {x c Co(T)+ Ix ~ ~ } ,

V xeCo(T).

y-

xC~

By Urysohn's Lemma, y is 1 on U and 0 on T \ U . Hence -1

u -

y (]0, ~ [ )

is open. Now let U be an exact set of T and take x E C0(T) such that U-

{x ~ 0}. Let

Y := V ( ~ / ~ (~lxl))

nEIN

e

do(T).

352

1. Banach Spaces

m

Then y is 1 on U and 0 on T \ U . Hence m

-1

u = y (]0, ~ [ ) is open.

m

The last assertion follows from the fact t h a t T is a Baire space. Example 1.7.2.14

( 0 ) Let T be a locally compact space. Take it 9 .h/lb(T)

and let F := Supp it 9 We identify .Mb(T) canonically with Co(T)' a) b)

# 9 Co(T) ~ iff every meager set of T is a # - n u l l set. -6 # 9 Co(T)~ ==vF = F .

c)

If T is Stonian and # 9 Co(T) ~ , then F is open and there is a unique

x 9 Co(T) with = ~.1,1,

I~1 < ~

In this case Ixl = eF. If in addition # is real, then x is also real and ,§

- ~§

,-

- ~-1,1

9

d) T is hyperstonian iff Co(T)co(T). is Hausdorff and in this case for every open nonempty set U of T

there is a ~ 9 Co(T)~_ with S u p p v

compact,

nonempty, and contained in U . a) Take # 9 Co(T) ~ . Let K be a nowhere dense compact set of T . Let 9= {x 9 Co(T) l e g <_ x} and denote by ~ the lower section filter of ~'. Since the infimum of 9v is 0, # ( K ) - lim f x,~ J

xd# - 0

o

Hence every meager set of T is a #-null set. Now suppose that every meager set of T is a #-null set. Let 9v be a downward directed nonempty set in Co(T)+ with infimum 0. Let ~ be its lower section filter and put

A :=

TI x infx/t/ 0 }

1.7 Ordered Banach spaces

353

._

Then A is a p-null set. Hence limx,~f

xd# = p(A) = 0

so that p 9 Co(T) ~ b) Since F \ F

is nowhere dense, it is, by a), a p-null set. Hence F\F

=0,

F=F

c) By b), F is open. There is a Borel function f on T such that /~-

fl~l,

Ifl--eF.

By Proposition 1.7.2.13 a =a b, there is an x 9 C ( T ) for which {x # f} is meager. By a), x=f

p-a.e.,

so that = x-I,I

9

Moreover, since T is a Baire space, IX I =

eF

9

The other assertions are easy to see. d) Let Fo denote the closure of

U uECo(T)

Suppu

.

~r

Assume that Fo :/: T . Then there is an x 9 Co(T)\{0} such that Supp x c T \ F o . We get xdu - 0

for all u C C0(T) ~ , i.e. Co(T)co(T),~ is not Hausdorff. Assume F0 = T . Take x C Co(T)\{0}. There is a u C Co(T)~+ with Suppx N S u p p u ~ 0 .

354

1. Banach Spaces

Then -2.u C Co(T)" and

f x d(~.,) = f lx ~du # O . Hence Co(T)co(Ty is Hausdorff. We prove now the last assertion. We may assume that U is open and compact. Let u C Co(T)~+ with U N Suppu r 0 . Then # := eu.u has the desired properties. Remark.

I

Co(T) '~ comprises precisely the order continuous elements of Co(T)'.

E x a m p l e 1.7.2.15

Let T be a discrete topological space and fiT its Stone-

Cech compactification. Given x C f l ( T ) and y C C(flT), define ~ " C(flT)

> IK,

Y'

)E

x(t)y(t)

tcT

and ~'~.I(T)

> IK,

x,

>E x ( t ) y ( t )

.

tET

a) 2" C C(flT)" for ev,:ry x E ~1 (T) and the map

is an isometry of ordered Banach spaces. b) fiT is a hyperstonian space. c) ~ E ~I(T)~ _ ~I(T), for every y C C(flT) and the map C(flT)

~ el(T) ',

y,

~ff

is an isometry of ordered Banach spaces. Given y c t ~ ( T ) , let fly denote the continuous extension of y to fiT. Then

e~(T) - - , C(gT),

y,

~&

is an isometry of ordered Banach spaces, a) and c) follow from Example 1.7.2.9 d) (and Example 1.7.2.8 d) and the last assertion of Proposition 1.7.2.5). By the above remark, C(flT) is order complete and so fiT is a Stonian space (Proposition 1.7.2.13 d ~ a). By a), fiT is a hyperstonian space.

I

1.7 Ordered Banach spaces

355

1 . 7 . 2 . 1 6 ( 0 ) Let T be a hyperstonian locally compact space, tl the (order complete) lattice of clopen sets of T , and 91 the (order complete)

Example

lattice of bands of Co(T)" (where we identify Co(T)' canonically with 3rib(T)). Given U 9 tl, put U--

(# 9 g0(T)'lSupp#

c U} .

Then U E 9l for every U C tl and the map 11

>91,

Ur

>U

is a lattice isomorphism. It is easy to see that U C 9I for every U C t/. The above map is injective since T is hyperstonian. Let N" E 91. We define

V "- U Supp # ,

U'-V

t~CN N

V is an open set (Example 1.7.2.14 c)), so U c t/. Take # E U. We want to show t h a t # c Af. For this we may assume ~hat Supp # is compact. Since

U \ V is nowhere dense, we may assume moreover t h a t Supp # is contained in V (Example 1.7.2.14 a) ). But then there is an L, C A f such t h a t Supp # c Supp u . From this relation it follows that # is absolutely continuous with respect to u, so # E N ' . Hence U C N ' . The converse inclusion is trivial. Thus U = Af and the map ~/

>9I,

U

>U

is bijective. It is easy to see t h a t it is even a lattice isomorphism.

I

If T is an infinite set, then the compact space {0, 1} T is

Example 1.7.2.17

not a-Stonian. For t C T , define :h'{0,1} T

>{0,1},

.

-

f,

>f(t),

356

1. Banach Spaces

Then Ut is a clopen set of (0, 1} T for every t E T . Let

(tn)neIN be

an injective sequence in T . For n E IN, put n--1

k=l

Then Vn is a clopen set for every n E IN. Hence

v"-U ~, nEIN

v".- U v~~ nEIN

are exact sets. Set

A'-

{tn i n E ]N).

Then eA belongs to the closures of both V' and V " . Since V' and V" are disjoint, their closures are not open.

I

357

Name Index

Name Index Alaoglu, L. Arens, R.F. Arzelg, C. Ascoli, G.

1.2.8.1, 1.5.2.10 1.1.2.16 1.1.2.14, 1.1.2.16

Banach, S.

1.1.1.2, 1.2.8.2, 1.3.1.2, 1.3.2, 1.3.3.1, 1.3.4.1, 1.3.4.10, 1.4.1.2, 1.4.2.3, 1.4.2.19 Bourbaki, N. 1.2.8.1 Branges, L. de 1.3.5.14 Cauchy, A. 1.3.10.6 Dedekind, R. 1.7.2.1 Dieudonn4, J. 1.2.8.2 Dworetzky, A. 1.1.6.14 Eberlein, W.F. 1.3.7.15 Fr4chet, M. 1.1.1.2, 1.1.2.13 Gelfand, I.M. 1.4.1.9 Goldstine, H.H. 1.3.6.8 Gowers, W.T. 1.2.1.12 Grothendieck, A. 1.6.1.1 Hahn, H. 1.1.1.2, 1.2.1.3, 1.3.3.1, 1.3.6.1, 1.3.6.3, 1.3.8.1, 1.4.1.4 Helly, E. 1.1.1.2, 1.3.3.13 James, R.C. 1.3.8.1 Kojima, ?? 1.2.3.11 Kolmogoroff, A. 1.1.1.2 Kottman, C.A. E 1.3.5 Krein, M.G. 1.3.1.10, 1.3.7.3 Laurent, P.A. 1.3.10.8 Lindenstrauss, J. 1.2.5.13 Liouville, J. 1.3.10.6 Mackey, G.W. 1.3.7.2 Milman, D.P. 1.3.1.10 Minkowski, H. 1.1.1.2, 1.1.3.4 Murray, F.J. 1.2.5.8 Neumann, J. von 1.1.1.2 Pettis, P.J. 1.3.8.4, 1.3.8.5 Phillips, R.S. 1.2.5.14, E 1.3.3

358

Riesz, F. 1.1.1.2, 1.1.4.4, 1.2.1.1, 1.2.2.5 Rogers, C.A. 1.1.6.14 Schmidt, E 1.1.1.2 Schur, I. 1.2.3.11, 1.2.3.12, 1.3.6.11 Sierpiflski, W. 1.1.2.17 Smulian, V. 1.3.7.3, 1.3.7.15 Steinhaus, H.D. 1.4.1.2 Stone, M.H. 1.3.4.10, 1.3.5.16 Toeplitz, O. 1.2.3.4 Weierstrass, K. 1.3.5.16

Subject Index

359

Subject Index NT means Notation and Terminology

(,4, B, (:)-multiplication

1.5.1.1

absolute value of a number

1.1.1.1

absolute value of a measure

NT

absolutely convex

1.2.7.1

absolutely convex closed hull absolutely convex hull

1.2.7.6

1.2.7.4

absolutely summable family additive group

1.1.6.9

NT

adherence, point of adherent point

NT NT

algebraic dimension algebraic dual

1.1.2.18 1.1.1.1

algebraic isomorphism, associated analytic function

1.2.4.6

1.3.10.1

Arens multiplication, left

1.5.2.10

Arens multiplication, right

1.5.2.10

associated algebraic isomorphism Baire function

1.2.4.6

1.7.2.12

Baire set

1.7.2.12

ball, unit

1.1.1.2

Banach categories, functor of

1.5.2.1

Banach categories, functor of unital Banach category

1.5.1.1

Banach category, unital Banach space

1.5.1.1

1.1.1.2

Banach space, complex

1.1.1.2

Banach space, ordered Banach space, real Banach system

1.7.1.4 1.1.1.2

1.5.1.1

Banach system, bidual of a Banach system, dual of a Banach systems, isometric band

1.7.2.1

1.5.1.9 1.5.1.9 1.5.2.1

1.5.2.1

360

bidual of a Banach system

1.5.1.9

bidual of a normed space bijective

1.3.6.1

NT

bilinear map

1.2.9.1

bitranspose

1.3.6.15

bound, lower

1.7.2.1

bound, upper

1.7.2.1

bounded map

1.1.1.2

bounded operator

1.2.1.3

bounded operator, lower bounded sequence bounded set

1.2.1.18

1.1.1.2

1.1.1.2 1.4.1.2

boundedness, principle of uniform canonical metric of a normed space canonical norm of s

F)

1.1.1.2

1.2.1.9

canonical projection of the tridual of E cardinal number

cardinality, topological

NT

carrier of a function

NT

carrier of a Radon measure category, Banach

NT

1.5.1.1

characteristic function of a set class

1.3.6.19

NT

1.1.2.1

NT

closed graph theorem

1.4.2.19 1.1.5.5

closed vector subspace generate by codimension codomain

1.2.4.1 NT

cokernel of a linear map compact, relatively

1.2.4.5 1.1.2.9

compatible, simultaneously

1.5.1.1

compatible (left and right) multiplications complement of a subspace

1.2.5.3

complemented subspace

1.2.5.3

complete, order

1.7.2.1

complete norm

1.1.1.2

complete normed space complete ordered set

1.1.1.2 1.7.2.1

completion of a normed space

1.3.9.1

1.5.1.1

Subject Index

361

complex Banach space

1.1.1.2

complex normed space

1.1.1.2

composition of functors

1.5.2.1

composition of maps cone

NT

1.3.7.4

cone, sharp

1.3.7.4

conjugate exponent of

1.2.2.1

conjugate exponents

1.2.2.1

conjugate exponents, weakly conjugate linear map

1.2.2.1

1.3.7.10

conjugate number

1.1.1.1

continuous, order

1.7.2.3

convergence, radius of convex

1.1.6.22

1.2.7.1

convex, absolutely

1.2.7.1

convex closed hull

1.2.7.6

convex closed hull, absolutely convex hull

1.2.7.6

1.2.7.4

convex hull, absolutely derivative

1.2.7.4

1.1.6.24

differentiable

1.1.6.24

dimension, algebraic Dirac measure direct sum

1.1.2.18

1.2.7.14 1.2.5.3

directed, downward

1.1.6.1

directed, upward

1.1.6.1

disjoint family of sets

1.2.3.9

distance of a point from a set domain

1.1.4.1

NT

downward directed dual, algebraic

1.1.6.1 1.1.1.1

dual of a Banach system

1.5.1.9

dual of a normed space dual space

1.2.1.3

1.3.1.11

equicontinuous equivalence class

1.1.2.14 NT

equivalence class of a point equivalence relation

NT

NT

362

equivalent norms

1.1.1.2

Euclidean norm evaluation

1.1.5.2 1.2.1.8

evaluation functor

1.5.2.1

evaluation operator of a normed space exact set

exponents, conjugate

1.2.2.1

exponents, weakly conjugate extreme point

1.2.2.1

1.2.7.9

face of a convex set

1.2.7.9

factorization of a linear map family

1.2.4.6

NT

family, absolutely summable family, sum of a

1.1.6.9

1.1.6.2

family, summable

1.1.6.2

family of sets, disjoint

1.2.3.9

filter, lower section

1.1.6.1

filter, upper section

1.1.6.1

filter of cofinite subsets finite-dimensional free ultrafilter function

NT 1.1.2.18

NT NT

function, Baire

1.7.2.12

function, step functor

NT

1.5.2.1

functor, identity

1.5.2.1

functor, inclusion

1.5.2.16

functor, isometric

1.5.2.1

functor, quotient

1.5.2.17

functor, transpose of a

1.5.2.3

functor of (unital) Banach categories functor of (unital) A-categories functor of (left, right) A-modules functors, composition of graph

1.3.6.3

1.7.2.12

NT,

groups, additive

1.4.2.18 NT

Hahn-Banach Theorem HSlder inequality

1.5.2.1

1.2.2.5

1.3.3.1

1.5.2.1 1.5.2.1 1.5.2.1

Subject Index

363

hyperstonian space

1.7.2.12

identity functor identity map

1.5.2.1 NT

identity operator iff

1.2.1.3

NT

image of a linear map ~maginary part inclusion functor

1.5.2.16

inclusion map

NT

induced norm infimum

1.2.4.5

1.1.1.1

1.1.1.2 1.7.2.1

infinite-dimensional infinite matrix mjective

1.1.2.18 1.2.3.1

NT

tuner multiplication interior point

1.5.1.1

NT

reverse of a bijective map reverse of a morphism

NT 1.5.1.6

reverse operators, principle of revertible

1.4.2.4

1.5.1.5

revertible, left

1.5.1.5

revertible, right

1.5.1.5

~sometric Banach systems ~sometric functor

1.5.2.1

1.5.2.1

~sometric normed spaces

1.2.1.12

lsometry of normed space

1.2.1.12

~somorphic normed spaces

1.2.1.12

~somorphism associated to a linear map, algebraic isomorphism of normed spaces kernel of a linear map

1.2.4.5

Kronecker's symbol lattice

1.2.2.6

1.7.2.1

lattice, vector

1.7.2.1

Laurent series

1.3.10.8, 1.3.10.9

left Arens multiplication left invertible

1.5.1.5

left multiplication left shift

1.2.1.12

1.5.1.1

1.2.2.9, E 1.2.11

1.5.2.10

1.2.4.6

364

left (unital) A-module, linear form

1.5.1.10

1.1.1.1

linear form, positive

1.7.1.9

linear map, conjugate lower bound

1.3.7.10

1.7.2.1

lower bounded operator lower section filter map

1.2.1.18 1.1.6.1

NT

map, bilinear

1.2.9.1

map, bounded

1.1.1.2

map, conjugate linear map, identity

1.3.7.10

NT

map, inclusion

NT

map, inverse of a bijective map, nuclear

NT

1.6.1.1

map, quotient

1.2.4.1

maps, composition of

NT

matrix, infinite

1.2.3.1

measure, Dirac

1.2.7.14

measure, Radon

NT

metric of a normed space, canonical modulo

1.1.1.2

NT

morphism

1.5.1.1

morphism, inverse of a

1.5.1.6

multiplication, (A, B, C) -

1.5.1.1

multiplication, compatible (left and right) multiplication, inner multiplication, left

1.5.1.1 1.5.1.1

multiplication, left (right) Arens multplication, right negative norm

1.5.1.1

1.7.1.1 1.1.1.2

norm, complete

1.1.1.2

norm, Eucliean

1.1.5.2

norm, induced

1.1.1.2

norm, quotient

1.2.4.2

norm, supremum norm of an operator

1.1.2.2, 1.1.5.2 1.2.1.3

1.5.2.10

1.5.1.1

Subject Index

norm of s

365

F ) , canonical

norm topology

1.2.1.9

1.1.1.2

normed space

1.1.1.2

normed space, bidual of a

1.3.6.1

normed space, complete

1.1.1.2

normed space, completion of a normed space, complex

1.3.9.1

1.1.1.2

normed space, ordered

1.7.1.4

normed space, real

1.1.1.2

normed spaces, isometric

1.2.1.12

normed spaces, isometry of

1.2.1.12

normed spaces, isomorphic

1.2.1.12

normed spaces, isomorphism of norms, equivalent nuclear map

1.6.1.1

number, cardinal

NT

number, ordinal

NT

object of a Banach system onto

1.5.1.1

NT

open mapping principle oprator

1.2.1.12

1.1.1.2

1.4.2.3

1.2.1.3

operator, bounded

1.2.1.3

operator, identity

1.2.1.3

operator, lower bounded

1.2.1.18

operators, principle of inverse order complete order continuous order of a pole

1.4.2.4

1.7.2.1 1.7.2.3 1.3.10.9

order summable

1.7.2.10

order a-complete

1.7.2.1

order a-continuous

1.7.2.3

ordered Banach space

1.7.1.4

ordered normed space

1.7.1.4

ordered set, complete

1.7.2.1

ordered set, totally

NT

ordered set, a-complete ordinal number

1.7.2.1 1.7.1.1

ordered vector space NT

366

partition of a set p--norm

NT

1.1.2.5, 1.1.5.2

point, adherent

NT

point, extreme

1.2.7.9

point, interior

NT

point of adherence polar

NT

1.3.5.1

pole (of order) positive

1.3.10.9

1.7.1.1

positive linear form

1.7.1.9

power series

1.1.6.22

precompact

1.1.2.9

predual of a Banach space prepolar

1.3.1.11

1.3.5.1

pretranspose of an operator principal part

1.3.4.9

1.3.10.8, 1.3.10.9

principle of inverse operators principle of open mapping

1.4.2.4 1.4.2.3

principle of uniform boundedness product of a family of sets projection

1.4.1.2

NT

1.2.5.7

projection of the tridual of E , canonical quotient functor quotient map

1.5.2.17 NT, 1.2.4.1

quotient norm

1.2.4.2

quotient space

1.2.4.2

quotient A-category

1.5.2.17

quotient A-module

1.5.2.17

radius of convergence

1.1.6.22

Radon measure

NT

range of values

NT

real Banach space

1.1.1.2

real normed space

1.1.1.2

real part

1.1.1.1

reflexive

1.3.8.1

relatively compact residue

1.1.2.9

1.3.10.8, 1.3.10.9

right Arens multiplication

1.5.2.10

1.3.6.19

Subject Index

367

right invertible

1.5.1.5

right multiplication right shift

1.5.1.1

1.2.2.9, E 1.2.11

right (unital) A-module, scalar

1.5.1.10

1.1.1.1

section filter, lower

1.1.6.1

section filter, upper

1.1.6.1

seminorm sequence

1.1.1.2 NT

series, Laurent

1.3.10.8, 1.3.10.9

series, power

1.1.6.22

set, Baire

1.7.2.12

set, bounded

1.1.1.2

set, complete ordered set, exact

1.7.2.1

1.7.2.12

set, partition of a

NT

set, totally ordered set, #-null

NT

NT

set, a-complete ordered sharp cone shift, left

1.7.2.1

1.3.7.4 1.2.2.9

shift, right

1.2.2.9

simultaneously compatible space, Banach

1.5.1.1

1.1.1.2

space, bidual of a normed space, complete normed

1.3.6.1 1.1.1.2

space, completion of a normed space, complex Banach

1.1.1.2

space, complex normed

1.1.1.2

space, dual

1.3.1.11

space, hyprstonian space, normed

1.7.2.12 1.1.1.2

space, ordered Banach

1.7.1.4

space, ordered normed

1.7.1.4

space, ordered vector space, quotient

1.7.1.1

1.2.4.2

space, real Banach

1.1.1.2

space, real normed

1.1.1.2

1.3.9.1

368

space, Stone

1.7.2.12

space, subspace of a normed space, vector

1.1.1.2

1.1.1.1

space, a - S t o n e

1.7.2.12

spaces, isometric normed

1.2.1.12

spaces, isometry of normed

1.2.1.12

spaces, isomorphic normed

1.2.1.12

spaces, isomorphism of normed step function

1.2.1.12

NT

Stone space

1.7.2.12

subspace, complemented

1.2.5.3

subspace generated by, closed vector subspace of a normed space sum, direct

1.2.5.3

sum of a family

1.1.6.2

summable, absolutely

1.1.6.9

summable, order

1.7.2.10

summable family

1.1.6.2

support of a function

NT

support of a Radon measure supremum

NT

1.7.2.1

supremum norm surjective

1.1.2.2, 1.1.5.2

NT

symbol, Kronecker's

1.2.2.6

Theorem of Alaoglu-Bourbaki Theorem of Banach

1.2.8.1

1.3.1.2

Theorem of Banach-Steinhaus Theorem of closed graph

1.4.1.2 1.4.2.19

Theorem of Hahn-Banach Theorem of Laurent

1.3.3.1

1.3.10.8

Theorem of Liouville

1.3.10.6

Theorem of Krein-Milman

1.3.1.10

Theorem of Krein-Smulian Theorem of Minkowski Theorem of Murray

1.3.7.3 1.1.3.4

1.2.5.8

Theorem of Weierstrass-Stone topological cardinality topology, norm

1.1.5.5

1.1.1.2

1.1.1.2

NT

1.3.5.16

Subject Index

369

topology, weak

1.3.6.9

totally ordered set

NT

transpose of a functor

1.5.2.3

transpose of an operator

1.3.4.1

transpose unital category of s

1.5.2.2

transposition functor of s triangle inequality

1.5.2.2

1.1.1.2

tridual of a Banach system

1.5.1.9

tridual of a normed space ultrafilter, free

1.3.6.1

NT

uniform boundedness, principle of unit

1.4.1.2

1.5.1.1, 1.5.1.4

unit ball 1.1.1.2 unit of an inner multiplication 1.5.1.1 unital Banach category 1.5.1.1 unital left A-module 1.5.1.10 unital right A-module 1.5.1.10 unital A-category 1.5.1.14 unital A-module 1.5.1.12 unital (A, A)-module 1.5.1.12 upper bound

1.7.2.1

upper section filter upward directed vector lattice vector space

1.1.6.1 1.1.6.1

1.7.2.1 1.1.1.1

weak topology 1.3.6.9 weakly conjugate exponents

1.2.2.1

A-categories, functor of (unital) A-category

A-category, quotient

1.5.2.17

A-category, unital A-module

1.5.2.1

1.5.1.14 1.5.1.14

1.5.1.12

A-module, left (right) A-module, quotient A-module, unital

1.5.1.10 1.5.2.17 1.5.1.12

A-module, unital left (right) 1.5.1.10 A-modules, functor of left (right) 1.5.2.1 A-subcategory 1.5.2.16

370

A-submodule 1.5.2.16 (A, A)-module 1.5.1.12 (A, A)-module, unital 1.5.1.12 #-null set NT a-complete order 1.7.2.1 a-complete ordered set 1.7.2.1 a-continuous, order 1.7.2.3 a-Stone space 1.7.2.12

Symbol Index

371

Symbol Index NT means Notation and Terminology m

A

NT

A

NT

~ ~ 1.3.5.1 A', A", A"' 1.5.1.9 AA, 1.3.6.9 a"x t 1.5.2.8 A/B

1.5.2.17

A+ B

1.2.4.1

A\B

NT

AAB AxB

NT

A+ z

1.2.4.1

NT

/3

1.1.2.4 NT

c

1.1.2.3

Co

1.1.2.3

c(T) co(T)

1.1.2.3

C(T)

1.1.2.4

1.1.2.3

C(T,E) Co(T) Card Coker

1.1.2.8 1.2.2.10 NT 1.2.4.5

dA

1.1.4.1

Det

NT

Dim

1.1.2.18

E'

1.2.1.3

E"

1.3.6.1

E'"

1.3.6.1

eA eT

1.1.2.1 1.1.2.1

et

1.1.2.1

eT

1.1.2.1

ET

1.1.2.1

372

E (T)

1.1.2.1

E~

1.7.2.3

E~

1.7.2.3

E+

1.7.1.1

E#

1.1.1.2

E+# E -5->F

1.7.1.4 1.5.1.1

E _5+ F

1.5.1.1

A

ElF

1.2.4.1

f'

1.1.6.24

~n

1.2.6.1 1.1.6.1

~I

/IS f(a, .) f(.,b) f(A) f(x)

NT

f-1

NT

NT NT NT NT

-1

f (B)

NT

-1

f (y)

NT

f ' X--+ Y NT f " X -+ Y . x ~> T(x) F[s,t] NT F[t] NT F 9G

1.2.5.3

{f = g}

NT

{f -r g}

NT

{f > a} gof -~A

im

NT NT, 1.5.2.1 1.7.2.3 1.1.1.1

Im

1.2.4.5

jE

1.3.6.3, 1.5.2.1

jEF IK

1.5.2.1 1.1.1.1

IK[.], IK[., .] N

k

1.2.3.1

1.1.1.1

NT

Symbol Index

373

U

k

1.2.3.1

Ker

1.2.4.5

s

1.2.1.3, 1.5.1.1

/2~

1.5.1.1

/2/

1.2.1.3

/21

1.6.1.1, 1.6.1.3

/2~

1.6.1.13

~P

1.1.2.5

t~P(T) t~~

1.1.2.5 1.1.2.3

t~~

1.1.2.3

t~~

1.1.2.2

g~(T)

1.1.2.2

gP'q(S, T)

1.2.3.2

go'q(S,T)

1.2.3.2 1.1.2.26 NT 1.1.3.3

A/Ib IN INn

1.1.2.1 ~/ Q

1.1.2.1 NT

IR

NT

IR

NT

re

1.1.1.1

Supp f Suppp

NT NT

u'

1.3.4.1

u"

1.3.6.15

U~(t) uT(t) X~ ,.., x -1 (x~)~cI

1.1.1.2 1.1.1.2 NT 1.5.1.6 NT

x'a" 1.5.2.8 xx', x'x 1.5.2.5 (x, x'}, (x', x> x" -t y", x" !- y"

1.2.1.3 1.5.2.10

374

{x I P(x)} NT {x e X IP(x)} NT (.,x')y 1.3.3.3 2Z

NT

z -t- A

1.2.4.1 1.1.1.1

Io~1

1.1.1.1 1.2.4.1

o~A

Ia,/3[, ]a,/3], [a,/3[, [a, ,3] A NT

6st

NT

1.2.2.6 1.2.7.14

5t

~(s, t)

1.2.2.6

]#1 I-I x~

NT NT

tEI

E x(t)

1.1.2.1

tET q

Xn

1.1.6.2

n--p

Y]xL

1.1.6.2

eel

1.3.3.3

~--~(., x'~)yL LEI oo

a'~x,~

1.1.6.22

n--0

~--]Sx~

1.7.2.10

~EI

1

1.2.1.3

1E + • \ (., .)

1.2.1.3, 1.5.1.5 1.2.4.1 NT NT 1.2.1.3

{-I.} NT {. = - } , {. :/: .}, {. > .} - (mod p) NT ~,l1.5.2.10

NT

Symbol Index

V,A

I]" II II-II~ II-I11 II-Iio I1 I1~ V, 3, 3!

375

1.7.2.1 1.1.1.2, 1.2.1.3

1.1.2.5 1.6.1.1 1.1.2.3 1.1.2.2

NT o NT, 1.5.2.1 G 1.2.5.3 [-,-], ]-,-[, [', [, ]-,]

NT

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